Multiple-scale analysis of transport phenomena in porous media

Transcription

THÈSE
En vue de l’obtention du
DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE
Délivré par l’Institut National Polytechnique de Toulouse
Spécialité : Hydrologie, Hydrochimie, Sol, Environnement
Présentée et soutenue par Yohan Davit
Le 3 décembre 2010
Sujet de la thèse :
Multiple-scale analysis of transport phenomena in porous media with biofilms
Analyse multi-échelles des phénomènes de transport dans des milieux poreux
colonisés par des biofilms
JURY
B. Goyeau
E. Paul
M. Quintard
S. Sauvage
R. Escudié
F. Delay
M. Gerino
G. Debenest
Professeur d'Université, EM2C
Professeur d'Université, LISBP
Directeur de Recherche, IMFT
Ingénieur de Recherche, ECOLAB
Chargé de Recherche, LBE
Professeur d'Université, HYDRASA
Professeur d'Université, ECOLAB
Maître de conférences, IMFT
Rapporteur
Rapporteur
Examinateur
Examinateur
Examinateur
Président du jury
Directeur de thèse
Directeur de thèse
Ecole doctorale : Sciences de l'Univers, de l'Environnement et de l'Espace
Unités de recherche : Institut de Mécanique des Fluides de Toulouse, ECOLAB
Directeurs de Thèse : G. Debenest, M. Gerino
Yohan Davit: Multiple-scale analysis of transport phenomena in porous media with
biofilms, Ph.D. thesis, © December 2010
ABSTRACT
This dissertation examines transport phenomena within porous media colonized
by biofilms. These sessile communities of microbes can develop within subsurface
soils or rocks, or the riverine hyporheic zone and can induce substantial modification
to mass and momentum transport dynamics. Biofilms also extensively alter the
chemical speciation within freshwater ecosystems, leading to the biodegradation of
many pollutants. Consequently, such systems have received a considerable amount
of attention from the ecological engineering point of view. Yet, research has been
severely limited by our incapacity to (1) directly observe the microorganisms within
real opaque porous structures and (2) assess for the complex multiple-scale behavior
of the phenomena. This thesis presents theoretical and experimental breakthroughs
that can be used to overcome these two difficulties.
An innovative strategy, based on X-ray computed microtomography, is devised
to obtain three-dimensional images of the spatial distribution of biofilms within
porous structures. This topological information can be used to study the response
of the biological entity to various physical, chemical and biological parameters at
the pore-scale. In addition, these images are obtained from relatively large volumes
and, hence, can also be used to determine the influence of biofilms on the solute
transport on a larger scale. For this purpose, the boundary-value-problems that
describe the pore-scale mass transport are volume averaged to obtain homogenized
Darcy-scale equations. Various models, along with their domains of validity, are
presented in the cases of passive and reactive transport. This thesis uses the volume
averaging technique, in conjunction with spatial moments analyses, to provide a
comprehensive macrotransport theory as well as a thorough study of the relationship
between the different models, especially between the two-equation and one-equation
models. A non-standard average plus perturbation decomposition is also presented to
obtain a one-equation model in the case of multiphasic transport with linear reaction
rates. Eventually, the connection between the three-dimensional images and the
theoretical multiple-scale analysis is established. This thesis also briefly illustrates
how the permeability can be calculated numerically by solving the so-called closure
problems from the three-dimensional X-ray images.
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RÉSUMÉ
Cette thèse se propose d’examiner les phénomènes de transport dans des milieux
poreux colonisés par des biofilms. Ces communautés sessiles se développent dans les
sols ou les roches souterraines, ou dans la zone hyporhéique des rivières et peuvent
influencer significativement le transport de masse et de quantité de mouvement. Les
biofilms modifient également très largement la spéciation chimique des éléments
présents dans le milieu, menant à la biodégradation de nombreux polluants. Par
conséquent, ces systèmes ont reçu une attention considérable en ingénierie environnementale. Pourtant, la recherche dans ce domaine a été très fortement limitée par
notre incapacité à (1) observer directement les microorganismes dans des milieux
poreux opaques réels et (2) prendre en compte la complexité multi-échelles des différents phénomènes. Cette thèse présente des avancées théoriques et expérimentales
permettant de surmonter ces deux difficultés.
Une nouvelle stratégie, basée sur la microtomographie à rayons X assistée par ordinateur, a été utilisée pour obtenir des images en trois dimensions de la distribution
spatiale du biofilm dans la structure poreuse. Ces informations topologiques peuvent
être utilisées pour étudier la réponse de l’entité biologique à différents paramètres
physiques, chimiques et biologiques à l’échelle du pore. De plus, ces images sont
obtenues sur des volumes relativement importants et peuvent donc être utilisées
pour étudier l’influence du biofilm sur le transport de solutés à plus grande échelle.
Pour cela, les problèmes aux conditions limites décrivant le transport de matière
à l’échelle du pore peuvent être moyennés en volume pour obtenir des équations
homogénéisées à l’échelle de Darcy. Différents modèles, ainsi que leurs domaines de
validité, sont présentés dans les cas réactifs et non-réactifs. Cette thèse se base sur la
technique de prise de moyenne volumique, en conjonction avec des analyses utilisant
les moments spatiaux, pour présenter une théorie complète de transport macroscopique ainsi qu’une analyse détaillée des relations entre les différents modèles, tout
particulièrement entre les modèles à une et deux équations. Une décomposition non
standard en terme de moyenne plus perturbation est présentée dans le but d’obtenir
un modèle à une équation dans le cas du transport multiphasique avec des taux de
réactions linéaires en fonction de la concentration. Finalement, la connexion entre
l’analyse théorique et l’imagerie en trois dimensions est établie. Cette thèse illustre
aussi brièvement comment la perméabilité peut être calculée numériquement en
résolvant des problèmes de fermeture à partir des images en trois dimensions.
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REMERCIEMENTS
Ce travail s’est déroulé en grande partie à l’institut de mécanique des fluides de
Toulouse (IMFT) dans le groupe d’étude des milieux poreux (GEMP), ainsi qu’au
laboratoire d’écologie fonctionnelle de Toulouse (ECOLAB). J’ai également eu la
chance de collaborer avec des chercheurs des universités de l’Oregon et du Montana.
J’aimerais saluer tous ceux qui, de près ou de loin, ont participé à l’élaboration et à la
réussite de ce projet.
En particulier, je tiens à remercier Michel Quintard, qui a été pour moi un modèle
d’excellence dans bien des domaines. Il fait partie de ces quelques personnes qui
ont marqué ma vie. Je n’oublierai pas les longues heures passées à griffonner des
moyennes volumiques, à discuter de la vie ou à explorer les gouffres de l’Aveyron.
Merci également à Guilhem et Brigitte Quintard pour leur amitié lors de ces week-end
spéléo, qui je l’espère vont pouvoir continuer.
Je voudrais exprimer mes plus sincères remerciements à Gérald Debenest pour
m’avoir toujours soutenu et pour avoir été parfait dans son rôle de directeur de thèse,
surtout dans les moments difficiles. Gérald m’a permis de m’épanouir scientifiquement, m’a fait confiance et n’a pas hésité a m’envoyer aux quatres coins du monde.
Merci!
I would like to thank Brian Wood for teaching me so much about science, for
making my trips to Corvallis so amazing and, more than anything, for his friendship. I
also want to emphasize how much I liked working with Gabriel Iltis, Ryan Armstrong,
Dorthe Wildenschild, James Connolly and Robin Guerlach. I hope our collaborations
will be long and fruitful. I was very lucky to work with such brilliant scientists. Toujours
dans les collaborations, j’ai eu énormément de plaisir à travailler avec Yoan Pechaud
et Etienne Paul au LISBP à l’INSA de Toulouse. Merci de m’avoir accueuilli dans votre
laboratoire, et pour tous les échanges que l’on a pu avoir.
Je tiens à exprimer toute ma gratitude à Frédéric Delay, Benoît Goyeau, Etienne
Paul, Renaud Escudié, Sabine Sauvage, Gérald Debenest, Michel Quintard et Magali
Gerino, pour avoir accepté de faire partie de mon jury de thèse et d’évaluer mon
travail.
Pour continuer, je vais faire un tour rapide de mes compagnons de route à ECOLAB
et à l’IMFT. Un énorme merci à tous mes amis GEMPiens: Stéphanie Veran, Alexandre
Lapene, Arnaud Pujol, Cyprien Soulaine, Yohann Le Gac, Marion Musielak, Florent
Hénon, Clément Louriou, Antoine Mallet, Pelforth et Ian Billanou. Je tiens également à
remercier Suzy Bernard, Marc Prat, Manuel Marcoux, Sylvie Lorthois, Rachid Ababou
et Lionel Lefur parce-que c’est un véritable plaisir de les croiser tous les jours. Je
salue également Sabine Sauvage, Karine Dedieu, Yvan Aspa, Frédéric Julien et Franck
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Gilbert que j’ai eu le plaisir de cotoyer à ECOLAB. C’est aussi l’occasion de remercier
tous mes amis et toute ma famille, qui ont participé à mon bien être en dehors du
laboratoire, en particulier Julia, Philippe, Sophie, Mij et évidemment mon père, ma
mère et Jean-Claude.
J’aimerais finir par la plus importante de tous, Janna. Merci pour ton amour, pour
ta patience et pour ton soutien.
CONTENTS
I
M ICROORGANISMS
IN POROUS MEDIA : A COMPLEX AND HIGHLY HETEROGE -
1
NEOUS FRAMEWORK
1
C ONTEXT: B IODEGRADATION IN POROUS MEDIA
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B IOFILMS
2.1 A bit of history . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Insights into the world of microbes: a picture tour . . . . .
2.3 The microbial modes of growth . . . . . . . . . . . . . . .
2.4 Dynamics of biofilm formation . . . . . . . . . . . . . . .
2.4.1 Cell transport and initial attachment . . . . . . . .
2.4.2 Growth and maturation . . . . . . . . . . . . . . .
2.4.3 Detachment . . . . . . . . . . . . . . . . . . . . .
2.5 Biofilm structure and composition . . . . . . . . . . . . .
2.5.1 General composition . . . . . . . . . . . . . . . . .
2.5.2 Extracellular polymeric substances (EPS) . . . . . .
2.5.3 Biofilm cells and their metabolism . . . . . . . . .
2.5.4 Water channels . . . . . . . . . . . . . . . . . . . .
2.6 Why do bacteria form biofilms ? . . . . . . . . . . . . . . .
2.7 Modeling biofilms . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Cellular-scale direct numerical simulations (DNS) .
2.7.2 Biofilm-scale empirical analysis . . . . . . . . . . .
2.7.3 Upscaling from the cell-scale to the biofilm-scale .
2.7.4 Advantages and disadvantages . . . . . . . . . . .
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B IOFILMS IN POROUS MEDIA
3.1 The significance of biofilms in porous media . . . . . . . . . . . . . .
3.2 Multiple-scale analysis of solute transport in porous media . . . . . .
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S COPE AND STRUCTURE OF THE THESIS
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II
T HREE - DIMENSIONAL IMAGING OF BIOFILMS IN POROUS MEDIA USING X- RAY
COMPUTED MICROTOMOGRAPHY
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1
I NTRODUCTION - I MPORTANCE OF DIRECT OBSERVATIONS
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M ATERIAL AND METHODS
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Contents
2.1
2.2
2.3
2.4
2.5
3
4
III
The porous models . . . . . . . . . . . .
Growing biofilms . . . . . . . . . . . . .
Contrast agent . . . . . . . . . . . . . .
Imaging protocols . . . . . . . . . . . .
2.4.1 Two-dimensional imaging . . . .
2.4.2 Three-dimensional imaging . . .
Data analysis . . . . . . . . . . . . . . .
2.5.1 Two-dimensional image analysis
2.5.2 Three-dimensional tomography
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R ESULTS
3.1 Two-dimensional experiments . . . . . . . . . . . . . . . . . .
3.2 Results of the 3-D tomography and discussion . . . . . . . . . .
3.2.1 Single polyamide bead . . . . . . . . . . . . . . . . . . .
3.2.2 Results for the polydisperse expanded polystyrene beads
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C ONCLUSION AND DISCUSSION
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M ODELING NON - REACTIVE NON - EQUILIBRIUM MASS TRANSPORT IN POROUS
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MEDIA
1
2
I NTRODUCTION
1.1 Microscopic description of the problem
1.2 Fully non-local models . . . . . . . . . .
1.3 Volume average definitions for this work
1.4 Two-equation models . . . . . . . . . .
1.5 One-equation models . . . . . . . . . .
1.6 Scope of this part . . . . . . . . . . . . .
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M ATHEMATICAL DEVELOPMENTS FOR THE MACROTRANSPORT THEORY
2.1 Two-equation ( MODELS B AND C ) . . . . . . . . . . . . . . . .
2.1.1 Volume averaging . . . . . . . . . . . . . . . . . . . .
2.1.2 Fluctuations equations . . . . . . . . . . . . . . . . .
2.1.3 Closure problems . . . . . . . . . . . . . . . . . . . .
2.1.4 Macroscopic equations Model B . . . . . . . . . . . .
2.1.5 Local Model C and conditions for time-locality . . . .
2.2 One-equation long-time behavior (M ODEL D ) . . . . . . . . .
2.2.1 Raw moments analysis . . . . . . . . . . . . . . . . . .
2.2.2 First centered moments and convergence . . . . . . .
2.2.3 Centered moments for the weighted average . . . . . .
2.2.4 Standardized moments, skewness and kurtosis . . . .
2.2.5 Constraints and convergence . . . . . . . . . . . . . .
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Contents
2.3
2.4
3
2.2.6 Numerical simulations . . . . . . . . . . .
2.2.7 Discussion . . . . . . . . . . . . . . . . .
Peculiar perturbation decomposition (M ODEL E)
Local mass equilibrium (M ODEL F) . . . . . . . .
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F ORMAL EQUIVALENCE BETWEEN Model D and E
3.1 Reminder of the expressions for the dispersion tensors . . . . .
3.1.1 One-equation time-asymptotic non-equilibrium model
3.1.2 One-equation special perturbation decomposition . . .
3.2 Mathematical development . . . . . . . . . . . . . . . . . . . .
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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N UMERICAL SIMULATIONS ON A SIMPLE EXAMPLE AND DOMAINS OF VALIDITY 121
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C ONCLUSION
IV
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M ODELING BIOLOGICALLY REACTIVE NON - EQUILIBRIUM MASS TRANSPORT
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IN POROUS MEDIA WITH BIOFILMS
1
2
I NTRODUCTION
1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Context . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 One-equation local mass equilibrium model
1.2.2 Multiple-continua models . . . . . . . . . .
1.2.3 A one-equation, non-equilibrium model . .
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U PSCALING
2.1 Microscopic equations . . . . . . . . . . . . . . . . . .
2.2 Averages definitions . . . . . . . . . . . . . . . . . . .
2.3 Averaging equations . . . . . . . . . . . . . . . . . . .
2.4 The macroscopic concentration in a multiphase system
2.5 Reaction term . . . . . . . . . . . . . . . . . . . . . . .
2.6 Non-closed macroscopic formulation . . . . . . . . . .
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3
C LOSURE
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3.1 Deviation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.2 Representation of the closure solution . . . . . . . . . . . . . . . . . 149
3.3 Closed macroscopic equation . . . . . . . . . . . . . . . . . . . . . . 151
4
N UMERICAL RESULTS
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4.1 Dispersion, velocity and reactive behavior . . . . . . . . . . . . . . . 156
4.2 Relationship with the local mass equilibrium model . . . . . . . . . . 160
4.3 Comparison with direct numerical simulation . . . . . . . . . . . . . 161
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Contents
4.4
4.3.1 Stationary analysis . . . . . . . . . . . . . . . . . . . . . . . . 162
4.3.2 Transient analysis . . . . . . . . . . . . . . . . . . . . . . . . 164
Conclusions concerning the numerical simulations . . . . . . . . . . 167
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D ISCUSSION AND CONCLUSIONS
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5.1 Relation to other works . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.2 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
V
C ONCLUSIONS , ONGOING WORK AND PERSPECTIVES
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1
C ONCLUSIONS
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2
O NGOING WORK AND PERSPECTIVES
2.1 Numerical perspectives . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Calculation of effective properties . . . . . . . . . . . . . . .
2.1.2 Pore-network modeling ? . . . . . . . . . . . . . . . . . . . .
2.1.3 Adaptative macrotransport calculations . . . . . . . . . . . .
2.2 Experimental perspectives . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Controlling the microbial species in order to study responses
to various environmental stresses . . . . . . . . . . . . . . . .
2.2.2 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Darcy-scale experiments . . . . . . . . . . . . . . . . . . . . .
2.3 Theoretical perspectives . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Adaptative macrotransport theory . . . . . . . . . . . . . . .
2.3.3 Mobile interfaces and growth/transport coupling . . . . . . .
2.3.4 And the other scales ? . . . . . . . . . . . . . . . . . . . . . .
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VI
R EFEREED PUBLICATIONS BY THE AUTHOR
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VII
APPENDIX
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A M ATHEMATICAL
DEVELOPMENT FOR THE TWO - EQUATION NON - REACTIVE
MODEL
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B M ATHEMATICAL DEVELOPMENT FOR THE TWO - EQUATION NON - CLOSED REAC TIVE MODEL
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C M ATHEMATICAL DEVELOPMENT FOR THE ONE - EQUATION PECULIAR DECOM POSITION MODEL
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CONTENTS
D I NTRODUCTION ET CONCLUSION EN FRANÇAIS
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D.1 Contexte - Biodégradation en milieux poreux . . . . . . . . . . . . . 207
D.2 Conclusions générales . . . . . . . . . . . . . . . . . . . . . . . . . . 209
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Part I
MICROORGANISMS IN POROUS MEDIA: A COMPLEX
A N D H I G H LY H E T E R O G E N E O U S F R A M E W O R K
1
C O N T E X T: B I O D E G R A D AT I O N I N P O R O U S M E D I A
Freshwater is a key component to life in general, and to human beings in particular.
Human activities already require a considerable amount of water and, obviously, the
consumption has the tendency to increase with population growth. Unfortunately,
unfrozen freshwater is relatively rare on Earth (about 1% of the total water) and
the quantity of contaminants released within aquatic ecosystems is also positively
correlated with human growth. Examples of one such pollution include:
• Agriculture which releases considerable quantities of organic and inorganic
pollutants such as insecticides, herbicides, nitrates or phosphates.
• Chemicals, such as solvents, detergents, heavy metals, oils which are present
on (unauthorized) dumping sites and can reach groundwater resources. For
instance, a huge quantity of European and American used electronic devices
end up in African countries [228, 185]. These are rarely treated and release
considerable amounts of extremely toxic compounds (lead, mercury, arsenic
and others). This electronic waste (e-waste) represents a significant pollution
and is a direct threat to human health [102].
• Medications are found in high concentrations within freshwater ecosystems.
For example, hormones present in birth-control pills, in particular estrogens,
deeply affect fish leading to feminization and intersex species [136].
As a direct consequence, water resources management requires specific attention.
Knowledge regarding the behavior of pollutants within these freshwater ecosystems
has become a matter of utmost importance. The contaminated area, that is, the original pollution site and the area that has been polluted through transport phenomena
must be clearly identified because they require special treatments. To this end, it is
fundamental to
1. understand the principal processes that are involved in these transport phenomena. One may think of this pollution as toxic molecules flowing through
rivers and lakes, but this is a very inaccurate picture. Freshwater is mostly
hidden within soil and subsurface zones. The transport of pollutants through
these geological formations involves complex phenomena including molecular
diffusion, convection, sorption, heterogeneous reactions and dispersion.
3
4
C O N T E X T : B I O D E G R A D AT I O N I N P O R O U S M E D I A
2. design and optimize techniques for the removal of the pollutants. For example,
organic pollutants can be degraded by a number of endogenous or exogenous
microorganisms. This phenomenon, usually termed biodegradation, refers to
the ability of microbes, especially bacteria, to modify the chemical speciation
of the elements present in the medium. Minute organisms can break toxic compounds to produce different chemical species. When the resulting chemicals
are inert or less toxic, the microbes can be directly used to purify the water.
For example, petroleum oils, containing toxic aromatic compounds, can be
degraded by hydrocarbonoclastic bacteria (HCB) [292]. On the other hand, the
biodegradation can also produce extremely toxic compounds and this must
be considered very carefully. For example, anaerobic biodegradation of perchloroethylene (PCE) can lead to a substantial production of trichloroethene
(TCE), dichloroethene (DCE) and vinyl chloride (VC).
To succeed in this, the transport processes require a quantitative description in terms
of mathematical modeling. Some basic approximations are often made in order to
undertake this description of the transport processes. In this thesis, pollutants are
treated as solutes. In essence, this means that they are always water-miscible, albeit
slightly and that the significant processes, in terms of transport, are those associated
with the miscible portion. For example, dense non-aqueous phase liquids (NAPLs)
form persistent blobs, trapped within the porous media, that dissolve slowly into
the water-phase. In addition, the solutes will be considered as tracers, potentially
non-conservative. This means that the contaminant is present in relatively small
concentrations and thus, does not significantly modify the density or the viscosity of
the fluid. This is generally the case for a large proportion of pollutants (for example
the NAPLs).
One also needs to characterize the aquifers themselves. For example, this can be
performed using specific non-reactive tracers. This solute can be introduced within
the aquatic medium, and breakthrough curves can be observed in wells spatially
distributed around the initial introduction area. The information collected from these
experiments can help in determining the direction of the flow and the general organization of the medium. However, proper interpretation of these data and theoretical
development of models often require topological information on a lower scale. For
example, X-ray computed microtomography can been used for this purpose, that is,
one can obtain a 3-D image of the porous structure at the microscale for a relatively
large volume.
The remainder of this part is organized as follows. First, we introduce the concept of
biofilm. This section, without being fully exhaustive, provides an up-to-date description of the biological entity and an overview of the fundamental principles regarding
the biofilm mode of growth. Secondly, we describe the significance of biofilms in the
context of porous media and particularly focus on the multiscale analysis of such
media. Finally, we describe the scope of this thesis, the global strategy and develop its
structure.
2
BIOFILMS
2.1
A B I T O F H I S TO RY
Microorganisms have been known to exist for centuries and yet, these are still triggering our scientific minds. Antoni van Leeuwenhoek (1632-1723) Fig (1a) was the first
to directly observe microbes through a homemade microscope Fig (1b). In 1674, he
describes, in letters to the Royal Society, the freshwater alga Spirogyra, and various
ciliated protozoa. In 1676, following the same line, he discusses the behavior of diverse “little animals by him observed in rain-well-sea and snow-water” [157]. He also
analyzed plaque bacteria collected from the surface of his teeth, in 1683. These were
the first insights into the world of microbes, and the first rebuttals against the theory
of spontaneous generation, synthesized by Aristotle and ultimately dismantled by
Louis Pasteur’s experiments in the 19th century. Antoni van Leeuwenhoek is now
widely referred to as the father of microbiology and bacteriology for his contribution
to our collective knowledge in these branches of science.
Global interest in microorganisms started later on, in the 19th century, in the
“fetid fever hospitals of Europe in the mid-1880s [...] when millions were dying of
plague and children were suffocating with diphtheria” [60]. In these morbid times, the
science of microbes imposed itself as a necessity and research focused on methodical
eradication more than on a general understanding of their behavior. Specific rules,
known as Koch’s postulates, were devised to establish a systematic causal connection
between a given microbe and a disease. The picture that one may formerly have had
of microbes was probably quite simple: weird shaped, free-floating minute organisms
that need to be eradicated.
One must wait until late 1920’s to early 1930’s to get a glimpse of the actual complexity and intricacy of these microorganisms. Pioneering works by Hilen [127], Hentschel
[126], Thomasson [253], Zobell [296], Winogradsky [281], Cholodny [50], Conn [56]
and Henrici [125] showed that microbes can grow together attached on surfaces.
They also noticed that, when attached, they seem to differ from their free-floating
planktonic form. Hilen (1923) described ”slimes on ships”, Hentschel (1925) and
Thomasson (1925) reported algae and diatoms behaviors on surfaces. The work undertaken by Zobell to describe sessile communities of bacteria was the real first step
toward a comprehensive description of their collective behavior on surfaces. His
5
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B I O FI L M S
(a)
(b)
Figure 1: (a) Antoni van Leeuwenhoek (1632-1723) and (b) Leeuwenhoek’s microscope.
contributions are detailed by Hilary M. Lappin-Scott in [152]. At the same time, engineers in wastewater treatment also realized that most of the microorganisms that
removed organic pollutants from water formed “slime” but also that, interestingly,
mixed species “show an obvious virility by far superior to that of pure culture” [15].
While the use of the word “virility” is opened to debate, the idea that these communities were in fact complex interacting consortia of microorganisms is a reality that
started to be encountered. After marine and freshwater organisms, the microbes of
the dental plaque have also received a lot of early attention [110, 60, 26, 267]. In the
1980s, the “slime” concept was put forth with the work by Characklis in [42] and by
Costerton in [132]. Characklis showed that “cities of microbes” colonizing surfaces are
extremely tenacious and resistant to biocides. Along this path, Geesey et al. in [106]
proved that most of the microorganisms in rivers live attached on rocks and Costerton
[132] unified the different researches. He proposed a comprehensive “biofilm” framework regarding the ability of microbes to adhere to surfaces and the advantages of
this ecological niche. Nowadays, biofilms have been shown to be extremely versatile
and found virtually everywhere a minimal amount of water and nutrients is available,
from your shower head to Yellowstone’s springs.
Further understanding of these sessile communities has come with two major
technical breakthroughs within the last decades:
• Knowledge regarding biofilms architecture has considerably improved with
the arrival of 3-D optical sectioning [155], that is, confocal laser scanning microscopy (CLSM is a depth selective microscopy technique, that can be used
to obtain three-dimensional information within non-opaque biological specimens, by scanning the medium point-by-point and then reconstructing it with a
computer). Previously, most studies used light microscopy to resolve spatially
cells embedded into a sticky matrix, which was still a fuzzy concept. Unfortunately, biofilm examinations based on light micrographs were found to be of
limited use because of their resolution and the two-dimensional projection
2.2 I N S I G H T S I N T O T H E W O R L D O F M I C R O B E S : A P I C T U R E T O U R
analysis. Some early attempts were performed using transmission electron
microscopy (TEM) and scanning electron microscopy (SEM) but the matrix
was severely affected by the process leading to dehydration artifacts [60, 156].
Nowadays, significant improvements have been made regarding the preparation of biofilms samples, for example using environmental scanning electron
microscopy, but CLSM remains the preferred technique, probably because it
captures a three-dimensional information and can measure a relatively large
volume.
• The development of molecular biology, proteomics and gene-sequence-based
analysis has also revolutionized our understanding of biofilms at the molecular
level. For example, various genes associated with the expression of the biofilm
phenotype can be functionally identified. In addition, microbes can be forced to
express a variety of exogenous proteins, through the incorporation of a specific
plasmid, to trigger different behaviors, or to improve imaging, for example,
using green fluorescent proteins (GFPs) [25, 214, 146].
All this collected information has lead to the realization that biofilms have a huge
sanitary, ecological and economic impact. Effects can be desirable (wastewater processes, bioremediation, industrial and drinking water treatment, sequestration of
CO2 ) or undesirable and, potentially, harmful (paper manufacture, contamination
in the food industry, medical infections, diseases, sustainability of water supply networks, microbially influenced corrosion (MIC) in pipelines, within heat exchangers
or on ships). The following are examples:
• Biofilms are responsible for approximately 65% of infections treated in the
developed world [117].
• 20% of corrosion inside heat exchangers is generated or influenced by biofilms
[99]. MIC can generate deeper corrosion and faster propagation than other
abiotic causes. In the United States, the daily cost of nuclear plants being idled
for maintenance of MIC within pipes is estimated at a million dollar a day
[34, 162].
• More than 99% of microbes cells in freshwater ecosystems live within biofilms
[59].
2.2
INSIGHTS INTO THE WORLD OF MICROBES: A PICTURE TOUR
The true naturalists among us have already explored the surfaces of our day-to-day
life objects, spending hours in contemplation of the animalcules that live there.
These explorations, using a simple camera, optical microscopy or more sophisticated
imaging techniques, are the cornerstone of the conception we have developed of
microorganisms. Hence, we start this introduction on biofilms with a picture tour,
7
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B I O FI L M S
which aimes to demonstrate at least a glimpse of their intricacy, their diversity and
even their beauty.
a) Grand prismatic spring
b) Spring biofilm
c) Spring biofilm
d) Mycobacterium smegmatis biofilm
Figure 2: Macroscopic manifestations of biofilms. Figure a) and b) source www.wikipedia.
com. Figure c) courtesy of S. Veran. Figure d) source www.microbelibrary.org.
Figure 2 shows examples of different manifestations of biofilms visible with the
naked eye. Figure (2a, b and c) illustrate the presence of biofilms around springs
where hot water and high concentrations of nutrients favorable the development of
microorganisms. In Figure (2a), the bacterial and algal mat, in the grand prismatic
spring in Yellowstone’s park (USA), is responsible for the beautiful orange, red and
yellow colors during summer (low chlorophyll concentrations). Figure (2b) was taken
near the grand prismatic spring and shows an outstanding amount of biofilms within
the water, green colored in the middle, where phototrophic organisms are in high concentration. Figure (2c) exhibits the same color range, but was taken in the geothermal
sources in Orakei Korako (New Zealand). In Figure (2d), mature Mycobacterium smegmatis biofilm (7 days old), a pellicle of microorganisms developing at the fluid-air
interface and primarily constituted of mycolic acids (waxes), makes strange patterns.
Figure 3 shows various scanning electron micrographs, exhibiting amazing details
on a nanometer scale. Figure (3a) is a colored scanning electron micrograph of an
acidophilic bacterial biofilm from a sample collected in a 75-m-deep borehole located
at the Kempton Mine complex in Maryland. In Figure (3b), Bacillus subtilis cells grow
2.2 I N S I G H T S I N T O T H E W O R L D O F M I C R O B E S : A P I C T U R E T O U R
a) Acidophilic bacterial biofilm
b) Bacillus subtilis
c) Bacterial biofilm
d) Salmonella typhimurium (red)
Figure 3: Microscale images (scanning electron micrographs). Image a) (G. Bowles, P. Piccoli,
P. Candela, M. Seufer, and S. K. Lower), cover photograph (2008) of Applied and
environmental microbiology. Image b) (B. Hatton and J. Aizenberg) source Wyss
Institute website. Image c) (P. Gunning, Smith and Nephew) source www.fei.com.
Image d) source www.ozonefreshsystems.com.
closely packed in parallel chains. In Figure (3c), various channels are visible within
the bacterial biofilm structure, growing on a micro-fibrous material and Figure (3d)
shows Salmonella typhimurium invading cultured human cells.
Figure 4 shows a strain of Shewanella oneidensis forming a biofilm under different
nutrient conditions. Confocal laser scanning microscopy is, nowadays, the most
utilized, and probably the most powerful, method to image biofilms on plane surfaces,
as it reveals fundamental structural details such as mushroom like colonies Figure
(4a).
Figure 5 represents a biofilm of Streptococcus mutans growing on a plane surface,
imaged using a new confocal technique (CRM) with an acquisition every 12 hours.
Such images give a really good idea of the heterogeneities inherent to the biofilm
lifestyle and make it possible to collect non-destructive precise dynamic information
about the growth of biofilms within microfluidic devices.
Very different types of microorganisms (bacteria, fungi, algae, archaea in interaction with protists) are known to produce biofilms. For instance, fungal biofilms
can develop on medical implants and can cause severe complications. However,
9
10
B I O FI L M S
Shewanella oneidensis
Figure 4: Confocal microscopy images of biofilms growth in different environmental conditions [169].
one may have noticed that most of the above microscopic figures focus on bacterial
biofilms. In a more general way, scientific interest has mainly focused on bacterial
biofilms. The reasons this are diverse. First, bacteria are the most abundant organisms
on Earth, meaning that, most of the time, they are the predominant species within
natural and engineered biofilms. Secondly, bacteria are probably the most versatile
microorganisms, and, hence, are particularly interesting for engineering applications
or biomimicry. Finally, bacteria are very ancient organisms that appeared on Earth
approximately 3.25 billion years ago (in comparison to Earth’s formation 4.6 billion
years ago) and thus, have played a very special role in the evolution of life.
2.3
THE MICROBIAL MODES OF GROW TH
Historically, the planktonic (lifestyle 1), or free-floating, phenotype has been the most
studied vegetative mode of bacterial development. For example, Escherichia Coli, a
Gram-negative bacterium (Gram staining is a procedure used to discriminate between
structural differences of bacterial cell walls. Gram-positive cell walls typically lack the
outer membrane found in Gram-negative bacteria) usually part of the mammals intestinal biota, has received a lot of attention and often serves as a model of flagellated
bacteria. Planktonic microbes play a crucial role in the elaboration of antibiotics
and biocides, as these are usually designed to be effective against the free-floating
phenotype.
As a a dormant non-reproductive response to tough environmental conditions,
some Gram-positive bacteria can also exhibit the bacterial spore, or endospore phe-
2.3 T H E M I C R O B I A L M O D E S O F G R O W T H
Streptococcus mutans
Figure 5: Microscale three-dimensional image obtained using confocal reflection microscopy
(CRM) [293].
notype (lifestyle 2). The cell coats itself with a thick protective outer wall, and goes
back to its normal vegetative state when conditions are favorable.
Even though lifestyles 1 and 2 have been examined for decades, the ultramicrobacterial or starvation form (lifestyle 3) is much less known. The realization that bacteria
could express such a phenotype is a recent discovery [141], their extremely small sizes
leading to difficulties of observation. Nowadays, ultramicrobacteria (UMB) are recognized as a fundamental dormant bacterial state and have been found in groundwater
as deep as 1500 m below the Earth’s surface and in the abyssal areas of the oceans
[60]. Similarly to lifestyle 2, it activates in response to a variety of environmental
stresses, such as starvation. Bacteria exhibit a huge amount of physiological mod-
11
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B I O FI L M S
ifications and can become significantly smaller, with a diameter of approximately
0.2 micrometer and a volume of about 0.02 cubic micrometer, than their normal
size (not to be confused with nanobacteria or nanobes which have an even smaller
normal vegetative size, i.e., 0.02 to 0.128 micrometer in diameter [265]). Remarkably,
when exposed to favorable conditions and nutrients, the UMB can also return to
their normal vegetative state [66]. Engineers have used these properties to develop
biobarriers technology that can contain pollutants within a restricted area (see Figure
6 adapted from [60]) [90].
Figure 6: Biobarrier process.
Biofilm (lifestyle 4) is the predominant mode of growth of bacteria. Evidence that
biofilm formation is an extremely ancient mechanism, that is, ~3.25 billions years
ago, has been reported in [217, 272, 215]. One may have trouble finding a consensus
2.4 D Y N A M I C S O F B I O FI L M F O R M AT I O N
of how to define biofilms as our conception of them is changing on a daily basis.
Here, we will adopt the definition given by Donlan and Costerton in [213], “the new
definition of biofilm is a microbially derived sessile community characterized by
cells that are irreversibly attached to a substratum or interface or to each other, are
embedded in a matrix of extracellular polymeric substances that they have produced,
and exhibit an altered phenotype with respect to growth rate and gene transcription”.
In addition to the “simple” definition, that is, microbes aggregated on a surface
that live embedded within extracellular polymeric substances (EPS), Donlan and
Costerton also consider the physiological and genetic modifications associated with
the biofilm phenotype [108, 109]. The proteins produced (genes expressed) by cells
within biofilms differ fundamentally from those produced by planktonic cells of the
same strain [225, 226, 3]. A very important consequence, discussed in Section 2.5.3,
of these profound modifications is that biofilms cells develop resistance to antibiotics
and biocides [239].
2.4
D Y N A M I C S O F B I O FI L M F O R M AT I O N
The formation of biofilms can be broken down into five different steps that are
summarized in Figure 7. This section proposes to follow the path of a free-floating
microorganism cell drifting within a fluid (e.g. water in a river) and to qualitatively
describe the physical, chemical and biological processes that lead to the formation of
a biofilm.
2.4.1 Cell transport and initial attachment
The journey of our minute organism can start within the water column of the Garonne,
a French river crossing Toulouse near the Institut de Mécanique des Fluides. The cell
will invariably exhibit a certain amount of movement that can look pretty chaotic, if
the cell is small and the river is quiet, or will follow the flow, especially during spring
when the flow rates are relatively large. Various processes are responsible for these
transport phenomena; which can be broken down into
1. a passive component: effects of local hydrodynamics, Brownian movements
and sedimentation [112].
2. an active component: motility of the cells. For example, microbes can get away
from toxic solutes or head toward nutrients. These motile behaviors include
chemotaxis, phototaxis and magnetotaxis [164, 101]. The mechanisms responsible for this movement are species-specific. Bacteria can use flagella for swimming through water; bacterial gliding and twitching for crossing surfaces; and
buoyancy modifications for vertical motion [16].
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B I O FI L M S
Figure 7: Schematics of biofilm formation.
At some point along its path, the cell will eventually reach the vicinities of the sediments bed that delineates the river channel. When it is close enough, the cell can
undertake reversible adsorption, often described in terms of DLVO theory (DLVO
[Derjaguin-Landau-Verwey-Overbeek] theory describes surface interactions of charged
colloidal particles (nanometer to micrometer sized) by taking into account van der
Waals attraction and electrostatic, double layer, repulsion forces), to a solid surface.
Various physical and chemical factors play a role in bacterial adherence. For example,
2.4 D Y N A M I C S O F B I O FI L M F O R M AT I O N
surface microtopography, flow rates, local pressures, substrate hydrophobicity and
electric charge, pH, salt concentration, type of organic compounds, time and temperature contact [135, 236, 297, 128, 61, 38, 261]. However, part of the process remains
biological, that is, a “decision” of the cell is involved. For example, initial attachment
can trigger specific membrane proteins [20, 32], especially for adherence to other
living organisms, or direct planktonic production of “sticky” exopolysacharides.
Eventually, the second step of biofilms formation occurs. Some attached bacteria undergo physiological modifications (expression of the biofilm phenotype) that
initiate the formation of microcolonies; the other cells can still exhibit detachment
mechanisms (described in section 2.4.3). This expression of the biofilm is fundamental but yet, not fully understood. The stimuli involved in this decision making are
barely tackled [60, 168]. Several fundamental questions have no definitive response:
1. How do bacteria sense the surface ? A current hypothesis is that signal molecules
secreted by the microorganisms are in higher concentration in the vicinities of
the solid surface (because, on the surface, the normal component of the flux is
zero [204]).
2. How do other planktonic species interact with the microcolonies ? It is expected
that other species will join the biofilm to form mixed bacterial consortia, specifically at this early stage of the biofilm formation. The first reason is that it is
easier for the other cells to attach to the matrix than to the abiotic surface. In
addition, a well established mature biofilm might not be inclined to accept new
species, that is, mixing organisms after the initial stage is arguably more and
more difficult [60].
3. Biological attachment and detachment are known to be closely related to quorum sensing phenomena (e.g. : [75, 291]) (Quorum sensing designates a fundamental ability of microbes to communicate with each other by molecular
signaling) but what are the molecular mechanisms involved ?
2.4.2 Growth and maturation
This step corresponds to the cells’ multiplication and biofilm expansion on the basis
of the first microcolonies. Three-dimensional structures develop [267] and the matrix
starts to interact strongly with the fluid. Then comes the maturation step. During this
stage, variations of the biovolume and of the biomass are relatively small as compared
to the growth step, but important structural modifications are triggered regarding
interspecies relationships
Various parameters influence the kinetics of growth and maturation. These include biological (such as which species is involved or predation) or physico-chemical
factors (such as local hydrodynamics [37], nutrient conditions [6], pH, presence of
biocides). There is a continuing debate within the scientific community regarding
15
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B I O FI L M S
the relative importance of the biological and the physical factors. One must keep in
mind that all aspects of the problem, even though they can seem very different, are
closely connected. Biofilms are in fact dynamic systems that continuously modify the
local physical, chemical and biological conditions surrounding them. These trigger
dynamic retroaction processes between the biota and the environmental conditions.
For example, Picioreanu (1998) [197] shows that local flux limitations can explain the
biological and physical heterogeneities within biofilms. Metabolically active cells,
initially chaotically distributed, consume and create various solutes and polymers,
modifying the chemical speciation of the elements present in the medium as well as
local hydrodynamics. Hence, gradients in concentrations and spatial heterogeneities
of the shear stress appear, influencing the development of ecological niches favorable
to different organisms [279, 289, 262, 94].
Figure 8: Modeling the formation of a two-species biofilm on a sphere in a mass transfer
limited regime using cellular automata. Source C. Picioreanu [195, 196].
2.4.3 Detachment
Detachment refers to the liberation of flocs or individual organisms from the biofilm.
There are various physical, biological and chemical processes responsible for these
detachments. It can occur when forces applied to the matrix exceed its internal
cohesion. This class of phenomena includes sloughing [213, 238], abrasion, predation
(see in Figure 9) or reduction of the cohesion by chemical attacks or enzymes. Erosion
refers to detachment induced by shear stress of the fluid on the matrix. Abrasion
occurs when solid particles flowing within the bulk fluid collide with the biofilm,
tearing off small flocs. An important feature of biofilms is that they display various
layers corresponding to different degrees of cohesion [105, 189]. Internal layers are
less exposed to hydrodynamics constraints, more dense and consequently more
difficult to remove than external layers.
The second class of detachment phenomena deals with active remobilization of
biofilms’ cells to colonize new niches [226] by “swarming/seeding dispersal” strategies or “clumping dispersal” [122]. Seeding dispersal corresponds to the release of
individual cells while clumping dispersal refers to the detachment of relatively large
2.5 B I O FI L M S T R U C T U R E A N D C O M P O S I T I O N
Figure 9: Example of protozoan (stentor) grazing biofilm [193].
aggregations of cells and extracellular polymeric substances. Hence, detachment
is a fundamental mechanism [262, 263] as (1) it partly controls the architecture of
biofilms, (2) it modifies the shape of the fluid/biofilm interface inducing changes in
mass and momentum transfer [238] and modifications of the adhesion properties,
rugosity for example, that facilitate the recruitment of new species [251] and (3) it
interferes with the proportions of the different species and their spatial distribution
[49, 176, 95, 85].
Even though the dynamics of biofilms formation are usually classified as in Figure 7,
there is actually no clear boundary between the different stages and this breakdown is
only a schematic description. For example, detachment phenomena are concomitant
to all other formation steps. For this reason, the artwork Figure 10 is outstanding
as, in a less schematic manner, it captures simultaneously the complexity of the
phenomena and the coexistence of all the processes.
2.5
B I O FI L M S T R U C T U R E A N D C O M P O S I T I O N
The architecture and the composition of biofilms are highly complex and diverse. For
example:
• It has been shown that the marine organism Serratia liquefaciens forms biofilms
in which the organism’s cells are “arranged into vertical stalks that bear rosettes
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B I O FI L M S
Figure 10: “Realistic” biofilm dynamics from [122].
of cells connected to other rosettes by long chains of cells and that each feature
of this architectural marvel is controlled by specific genes [149]” (from [60]).
• One clone of Pseudomonas aeruginosa (Pseudomonas aeruginosa is a Gramnegative, aerobic, rod-shaped bacterium and an opportunistic human pathogen)
forms “stumplike pedestals on colonized surfaces and that mobile cells of a
second clone crawl up the pedestals and form the ’caps’ of the mushrooms that
are such a prominent feature of biofilms formed by this organism [258]” (from
[60]).
The composition and the architecture of biofilms can significantly change from
one biofilm to another. However, some aspects of their global composition and
some patterns are specific to the biofilm phenotype. For example, well-fed biofilms
are often flat and poorly structured whereas low nutrient concentrations lead to
highly structured communities. The goal of this section of the thesis is to present the
common composition and the key structural elements of biofilms.
2.5.1 General composition
Component
% (mass) of matrix
Water
up to 97%
Microbial cells
2-5%
Polysaccharides
1-2%
Proteins
<1-2%
DNA and RNA
<1-2%
Ions
?
Inorganic
?
Table 1: General composition of biofilms, adapted from [246].
Following Sutherland (2001) [246], the general composition of biofilms can be
described as in Table 1. Most of the biofilm matrix (97% of the total mass) is actually
water. The water can be bound within the capsules of microbial cells or can exist as
a solvent whose physical properties such as viscosity are determined by the solutes
dissolved in it [246]. Extracellular polysaccharides, proteins, DNA and RNA can be
integrated into the notion of extracellular polymeric substance that is described in
the next section. Inorganic compounds such as minerals or clay particles can get
immobilized on the biofilm, depending on the environmental conditions.
2.5.2 Extracellular polymeric substances (EPS)
The EPS definition is in constant evolution, as new constituents, processes and functions are frequently discovered. Old definitions include: organic polymers that are
often responsible for the cohesion of the cells within biofilms and for their adhesion to substrates [72]. One significant further step has come with the unification
of the different concepts that are used to describe this matrix [154]. Nowadays, the
EPS is known to be composed of a complex mixture of macromolecules including
exopolysaccharides, proteins, DNA, RNA [245, 246] in addition to peptidoglycan,
lipids, phospholipids and other cell components. They (1) are usually thought as the
principal structuring component of the matrix, that is, the cement that holds the
microorganisms together and to the surface and also (2) play other functional roles
(Table 2). For example, alginate is a polyanion polysaccharide that is supposed to be
involved in Pseudomonas aeruginosa’s resistance to antibiotics [122].
The extracellular DNA present in biofilms has been thought for many years to be
only the result of cell lysis. However, it has been observed that DNA can be produced
and sent outside the cell in small membranes [182, 137]. More recently, it has been
shown that absence of this DNA can inhibit the biofilm growth [275] and that it can
exhibit a very specific structural organization [27]. The intricacy and functionality of
EPS remain to be fully understood, especially in natural environments. Most of the
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B I O FI L M S
EPS component
Role in biofilm
Neutral polysaccharides, amyloids
Structural components
Charged or hydrophobic polysaccharides
Ion exchange, sorption
Extracellular enzymes
Polymer degradation
Amphiphilic molecules
Interface interactions
Membrane vesicles
Export from cell
Lectins, nucleic acids
Specificity, recognition
Sorption
Genetic information
Structure
Bacterial refractory polymers
Electron donor or acceptor?
Various polymers
Source of C, N, P
Table 2: The EPS defined, adapted from [100].
work has focused on Pseudomonas aeruginosa, or on other laboratory species, but
natural biofilms seem particularly different [60, 100]. For instance, the production of
EPS in natural biofilms is dynamic and can follow cyclic patterns as demonstrated in
marine stromatolites [84] or can form honeycomb structures [89, 167, 252].
2.5.3 Biofilm cells and their metabolism
Bacterial metabolism englobes the ensemble of biochemical reactions that produce
the energy and solutes that are necessary for the cell to function. There is an incredible diversity of cells belonging to various classes. Presenting all these metabolisms
is far beyond the scope of this introduction. Here, we will focus on some important characteristics of bacteria and the modifications associated with the biofilm
phenotype.
Bacteria can be classified by their nutritional groups
following the carbon source, the kind of energy source, the electron donor and the
electron acceptor used for their development. Phototroph organisms take their energy
from sunlight, by photosynthesis, while chemotrophs use chemical reactions that
produce energy, mostly oxidation of electron donor at the expense of an electron
acceptor such as dioxygen or nitrates. Among chemotrophs, there are lithotrophs that
oxide inorganic compounds and organotrophs that use organic molecules. Eventually,
during the catabolism, adenosine triphosphate (ATP) is synthesized as the energy
source for the cell.
On the basis of the carbon source, bacteria are divided into heterotrophs, which
use organic carbon and autotrophs, which feed on inorganic carbon sources, mostly
on CO2 . Organisms can be further separated into aeroby if they use oxygen as the
electron acceptor or anaeroby if they do not. This classification has its limitations but
B A C T E R I A L M E TA B O L I S M
is often utilized. For example, bacteria also need nitrogen, potassium, oxygen and
minerals to form the amino-acids required to synthesized proteins; some bacteria
can be heterotrophs with respect to the carbon source but autotrophs in relation to
other basic compounds such as nitrogen.
Cells exhibit fundamental phenotypic modifications
associated with the biofilm lifestyle. The pattern of genes expression has been found
to differ by between 20% and 70% as compared to the corresponding planktonic cell
[22, 60]. A direct consequence of these modifications is that biofilm cells are more
resilient and exhibit resistance to antibiotics and biocides. It was first suggested that
biofilm acts as a barrier which prevents harmful components from reaching the cells.
However, while it certainly slows down the diffusion, there has been a substantial
amount of evidences that antibiotics can penetrate hundreds of micrometers into the
matrix within seconds [183, 243]. Another hypothesis is that biofilms’ cells lack the
enzymes that are usually targeted by antibiotics [111, 240, 3, 225, 226].
In addition, within biofilms, the microbial cells, whether from the same species or
from different species, are in close proximity to one another. This has various direct
consequences:
S P E C I FI C I T Y O F B I O FI L M S
• Metabolic symbiosis can appear. For example, some species can use solutes
produced by other species [5, 96] or different cells can exchange genetic material [103, 124]; specialization through stratification can occur in response to
various environmental conditions. For example, in aerobic conditions, aerobic
bacteria develop in surface of the biofilms while anaerobic ones tend to form
deeper within the biofilm, where oxygen has been already consumed by surface
organisms [165].
• On the contrary, competition for available nutrients is intense; antimicrobial
substances released by some cells, such as bacteriocins, microcins, antibiotics
or phage, have a good chance of successfully attacking and possibly destroying
neighboring heterologous cell types [133]; predator-prey dynamic relationships
affect organisms within the matrix [180, 187].
Complex networks of relationships develop between the cells, that eventually end up
forming evolutive microbial consortia, that continuously change.
2.5.4 Water channels
In many natural environments, biofilms form mushroom-like structures sprinkled
with water channels [179] in which convective flows have been identified [160, 241].
The channels act as nutrient suppliers for the deepest microorganisms [158, 82]. To
get a clear picture of these channels, one may imagine the way oxygen is carried out
in the human body. For example, brain cells cannot live more than a few seconds
21
22
B I O FI L M S
without oxygen, that is, the time for O2 to diffuse through several millimeters of
flesh. If the oxygen were to diffuse all the way from the lungs to our brain, it would
take hours, and cells would die. Hence, in multicellular organisms larger than a few
millimeters, veins convey the oxygen toward the cells until the critical millimeter
length scale is reached. In the biofilms, the situation is similar. In thick biofilms, if
the deepest cells were to wait for the nutrients to diffuse all the way through the
biofilm, most of them would die. Nutrient channels are thus necessary. To what extent
the formation of these channels is the result of an evolution of the microorganisms
toward that end or a consequence of the transport processes on a shorter time-scale
remains to be fully understood.
2.6
W H Y D O B A C T E R I A F O R M B I O FI L M S
?
Figure 11: Artistic interpretation of the four driving forces behind bacterial biofilm formation
[134].
The forces that drive bacteria to produce biofilms are closely connected to the theory of evolution. Bacteria tend to develop embedded within extracellular polymeric
substances because this lifestyle promotes survival and growth of the population.
That being said, the real question becomes: What are the physical, chemical and
biological phenomena that make the biofilm mode better than the planktonic one?
2.6 W H Y D O B A C T E R I A F O R M B I O FI L M S ?
Following Jefferson’s work [134], these phenomena can be classified according to four
different classes, as illustrated on Figure 11.
First of all, the biofilm form can develop as a defense mechanism in response to
various environmental stresses such as mechanical stress (shear or osmotic), UV
exposure, metal toxicity, acid exposure, dehydration and salinity, phagocytosis, iron
deprivation and several antibiotics and antimicrobial agents [122]. EPS is thought
to play a crucial role in the protection of the cells. For example, it might protect
organisms from being crushed by important mechanical stresses, reduce the quantity
of several toxic compounds that diffuse through the matrix and reach the deepest
organisms or simply slow down the diffusion allowing the bacterial immune system
to mount a specific attack. Evidence for such mechanisms includes the expression
of specific proteins that trigger the expression of EPS in response to environmental
stresses such as the upregulation of proU in E. Coli to adapt to transport or osmotic
stresses [204]. Another example is the creation of starved, stationary dormant zones
that seem to induce resistance in biofilm populations to antimicrobial agents. Arguably all biocides or antibiotics need at least some degree of metabolic activity in
order to be efficient and, hence, might not kill dormant bacteria [122].
The second class of phenomena deals with fighting host immune systems in order
to stay in a favorable habitat. For example, the human body, because of its adequate
temperature and high concentrations of nutrients, represents an extremely appealing
habitat for microorganisms. The counterpart is obviously that they will have to develop either in symbiosis with the host or to constantly struggle against its immune
system. From this perspective, the biofilm mode of growth is an excellent strategy to
remain fixed to the host. Specific microbial surface component recognizing adhesive
matrix molecules (MSCRAMMs) are often expressed and play a key role in initial
adherence of bacteria to host surfaces [194].
There are also various advantages associated with life in community
• A division of labor can develop. As previously discussed, some symbiotic relationships might occur. For example, myxobacteria can produce extra digestive enzymes as a community and, potentially, break down complex food substances. Communication is realized through the expression of auto-inducing
molecular signals, that is, as previously discussed, quorum sensing.
• Because of their proximity, biofilms cells can perform genes transfer. These processes may directly benefit to the bacteria, for example by exchanging antibiotic
resistance determinants [134].
• Some microorganisms might exhibit programmed cells death mechanisms, in
order to reduce to metabolic activity in low nutrients conditions, that is, the
biofilm might behave like a multicellular organism, although this is still a matter
of debate [134].
23
24
B I O FI L M S
• There are also evidences that the biofilm formation is the default mode of
growth of microorganisms. When bacteria sense an adequate surface, that is, a
free ecological niche, they develop the biofilm phenotype. For example, algC, a
gene required for alginate synthesis, in Pseudomonas aeruginosa is upregulated
within minutes of attachment [74]. Jefferson (2004) suggests that one may
have to wonder more about what drives bacteria to develop the planktonic
phenotype.
John Tyler Bonner would probably argue that size increase is the meaningful evolutionary concept that regroups all these advantages. In [28], he discusses the importance of size and scales on the evolution of Earth’s organisms and states that “size
is a prime mover in evolution”. Bonner also says that “there has been a selection for
larger organisms; under some circumstances size increase can confer advantages and
promote reproductive success [...]. If there is an advantage to increased size, one of
the simplest ways to achieve it is to become larger by becoming multicellular”. The
idea behind that is an ecological principle based on Darwinian evolution. There is a
perpetual competition for survival in natural habitats. All size levels, except for the
largest one, are occupied by some organisms. Biofilm could have developed originally as a result of cells aggregation (due to passive and maybe active phenomena)
and have survived as the top-size free niche promotes survival and growth of the
population.
2.7
M O D E L I N G B I O FI L M S
Developing mathematical models has become a matter of utmost importance in
scientific research in general. Direct observations are the cornerstone of scientific
knowledge. Yet, quantifiable understanding regarding the processes at play as well
as quantitative predictions cannot be achieved without mathematical analysis. This
is particularly true within the context of biofilms where numerous extremely complex processes are at play simultaneously. B. E. Rittmann (2007) [221] states that
mathematical modeling provides a quantitative relationship between the biofilm
components and the processes. “The components include the various microbial
types, extracellular polymeric substances (EPS), inert or dead biomass, the metabolic
substrates for all the microbial species, metabolic products from the reactions they
carry out, alkalinity, they hydrogen ion (for pH), and more. Processes include the
metabolic reactions, transport of solutes and biomass, acid/base reaction, precipitation/dissolution, and physical deformations due to forces acting on the biofilm. Many
of the processes, particularly the metabolic ones, are the services that the community
provides human society”. In a more general framework, mathematical modeling can
help in identifying pertinent physical, biological and chemical parameters as well as
discerning irrelevant ones. It also represents a unique predictive technique that can
provide informations regarding future behaviors. Hence, models have become a key
2.7 M O D E L I N G B I O FI L M S
component to provide responses to specific engineering applications or are used as a
research tool.
Along with the realization of this modeling necessity, a substantial amount of
theoretical works have been developed within the last decades. One critical aspect
of the problem, which is particularly important in this work, is that biofilms are
extremely heterogeneous. According to Bishop and Rittmann (1995), heterogeneity
may be defined as "spatial differences in any parameters we think is important". An
adapted list from Bishop and Rittmann (1995) [24], summarizes a few examples of
possible biofilm heterogeneities:
1. Geometrical heterogeneity: biofilm thickness, biofilm surface roughness, biofilm
porosity, substratum surface coverage with microbial biofilms.
2. Chemical heterogeneity: diversity of chemical solutes (nutrients, metabolic
products, inhibitors), pH variations, diversity of reactions (aerobic/anaerobic,
etc.).
3. Biological heterogeneity: microbial diversity of species and their spatial distribution, differences in activity (growing cells, EPS producing, dead cells, etc.).
4. Physical heterogeneity: biofilm density, biofilm permeability, biofilm viscoelasticity, viscosity, EPS properties, biofilm strength, solute concentration, solute diffusivity, presence of abiotic solids.
Length scales that are involved vary from nanometers (EPS polysaccharides) to millimeters or even centimeters (biofilm thickness) with an intermediate size of about
several micrometers (cells) (Figure 12). All of these spatial scales are pertinent when
working with biofilms. For example, let us consider the path of a nutrient molecule
that originally is in the bulk fluid. It might be submitted to diffusion and convection
within the water, then goes through the millimeters of EPS and eventually reaches the
membrane of a cell. Once there, it has to cross the interface using specific transmembrane proteins such as porins.
These heterogeneities are extremely difficult to model because of the broad range
of scales that are involved. One usually tries to avoid a purely multiscale description
of the various processes. The primary reason for this is that our computational resources are limited and molecular modeling over centimeters is nearly impossible.
Secondly, such models would contain an enormous amount of information that is
mostly irrelevant at the macroscale. Various strategies have emerged to model the
macroscopic behavior of biofilms, that is, to describe the phenomena in some averaged sense. Microscale modeling strategies as well as upscaling techniques and
empirical formulations are presented and compared within the next sections.
25
26
B I O FI L M S
Figure 12: Multiscale representation of biofilms (adapted from [138]).
2.7.1 Cellular-scale direct numerical simulations (DNS)
A cellular automaton (CA) model consists in an array of
small compartments, each in one of a finite number of states. For each time step
t+dt, a new generation is created as a function of the generation at time t, following
a given set of rules. For example, the creation of a new microorganism cell requires
that the set of rules specifies where to place it, as a function of the neighborhood
of the mother cell. The rules typically take into account processes such as substrate
uptake, metabolism, maintenance, cell division, transport and death. Although the
main idea is the same, the set of rules that govern the spreading of the biomass differs
from model to model [153]. Examples of early CA models include [104] and [280].
C E L L U L A R A U T O M ATA
I N D I V I D UA L - B A S E D M O D E L S
Individual-based models simulate individual cells,
rather than the behavior of a predefined grid (such as CA). All other aspects of the
simulation, and the processes that are modeled, are very similar to those of the
cellular automata. Advantages, disadvantages and problems that need to be tackled
are thoroughly discussed in [153].
2.7.2 Biofilm-scale empirical analysis
Empirical analysis remains the most common technique. The macroscopic laws are
guessed on the basis of the mathematical structure of the problem at the microscale,
in conjunction with experimental data. As an example of one such technique, we
will describe how cells transport is commonly described within the fluid phase. One
usually cannot follow the Lagrangian trajectory of every single cell in the medium, but
rather, assumes that these can be followed in terms of an homogenized concentration.
In addition, cells transport within the bulk fluid is usually described in terms of a mass
balanced equation in which it is assumed that fluxes are convective and diffusive.
This is based on the assumption that the attachment mechanisms can be uncoupled
from the transport processes, that is, that the characteristic times for attachment are
much longer than those associated to diffusive and convective transport. If such an
hypothesis is valid, then the mass-balance equation takes the form (e.g. [112])
∂wγ
+ ∇ · {(vγ + ζvs ) wγ } = ∇ · {kγ ∇wγ }
∂t
(2.1)
which is an heuristic formulation of the Fokker-Planck equations for particles (see
discussion in [212]). In Eq (2.1), wγ refers to the concentration of cells within the
water γ − phase, vγ is the pointwise Eulerian velocity field (solution of the NavierStokes equation) within the γ − phase and kγ is the diffusion coefficient of the cells
within the water. vs is the maximum theoretical velocity for the sedimentation of a
spherical particle in an infinite medium, that is,
vs =
(ρp − ρ) gd2p
18µ
(2.2)
where ρp is the buoyant density of the particle and ρ the one of the bulk fluid. dp is
the diameter of the sphere, µ the viscosity of the bulk fluid and g is the gravitational
acceleration. ζ is an empirical parameter such as 0 6 ζ 6 1, that takes into account,
among others, the grain surface rugosity and the sphericity of the particles (see
discussions in [123]).
2.7.3 Upscaling from the cell-scale to the biofilm-scale
The idea that biofilm models should be upscaled directly from the microscopic
scale has been brought forth previously by Wanner and Gujer (1986) [271]. One
can adopt the suggestions of Wanner et al. (1995) [269] by starting at the cellular
level, where the biofilm is a discontinuous multiphase medium [285, 286]. B. D. Wood
an co-workers performed such an analysis using the volume averaging with closure
techniques [287, 284]. These include upscaling reactive solute transport within the
27
28
B I O FI L M S
matrix, cellular growth and multispecies diffusion. In general, most of the work with
volume averaging has focused on local mass equilibrium conditions, that is, when
the gradients of concentrations within each phase are relatively small, the continuity
conditions at the interfaces can be extended to the bulk phases. Hence, all intrinsic
concentrations are equal and the mass transport can be described in terms of a single
mass transport equation.
2.7.4 Advantages and disadvantages
DNS
Advantages
Disadvantages
Realistic models
Time-consuming
Large amount of information
Limited to small sized volumes
No microscale validation
Empirical analysis
Simple laws
No connection macro/micro
No domain of validity
Upscaling
Connection macro/micro
Non-equilibrium conditions
Domains of validity
Non-linearity
Description of large volumes
Dynamic interfaces
Table 3: Advantages and disadvantages of the different models.
DNS (Direct Numerical Simulation) has been used to explore some physics of
biofilm formation. In particular, cellular automata represent a very particular framework, that has recently met a great success. It can be used for different purposes, for
example to study the influence of nutrient flux limitations within biofilms. It models
biofilms starting at the cellular-scale, and hence, is quite realistic and captures a lot
of information. On the other hand, these are often time-consuming simulations, that
are limited to a relatively small total volume. In addition, each cell receives a set of
basic rules that change from one model to another. The biofilm growth is obviously
very sensible to these rules, and validation is not easy to perform.
Empirical development of macroscopic laws often ends up with relatively simple
macroscopic equations that are convenient for data interpretation. However, upscaling techniques in general (for instance, homogenization, volume averaging, moments
matching techniques), and volume averaging in particular, (1) relates the microscopic
parameters to the observable macroscopic ones. As part of the process of volume averaging, the microscopic parameters that apply to these small scales can be explicitly
linked to their macroscopic counterparts and (2) volume averaging indicates under
what conditions the conservation equations are valid. The counterpart is the difficulty to treat non-equilibrium situations, non-linearity or dynamics of the interfaces
such as biofilm growth coupled with mass transfer. However, one must remember
that these disadvantages are purely technical and can be overcome. If an averaged
behavior exists, then upscaling techniques should be able to capture it. If it does not,
then one needs to use DNS.
29
3
BIOFILMS IN POROUS MEDIA
3.1
T H E S I G N I FI C A N C E O F B I O FI L M S I N P O R O U S M E D I A
? In freshwater ecosystems,
99% of the microorganisms live within periphytic biofilms [60]. All of us have experienced the predominance of this slippery form of life, falling unexpectedly in the water.
Yet, most of the bacteria remain hidden underneath the sediment surface. In fact, the
riverine hyporheic compartment (The hyporheic zone is the region beneath and lateral
to the stream bed, where flow rates are sufficient to mix shallow water with surface
water) has been identified as particularly active in terms of bio-chemical processes.
For example, this zone is known to play a key role in the self-purification properties of
the rivers and in the biodegradation of toxic compounds. One may then wonder why
microbes love living within these porous media so much and not just on the bottom
of the river ? Porous media actually represent an extremely favorable habitat in terms
of defensive capacity,
W H Y D O B I O FI L M S F O R M W I T H I N P O R O U S M E D I A
• Microorganisms are partly protected from shear stresses as the fluid velocity
within the water column is much larger than within the pore throats.
• Steric hindrance might also limit predation, that is, relatively large predators
are slowed down by the sediments. These have developed specific bioturbation
techniques but the predator-prey equilibrium might be considerably modified.
• The surface available for attachment is particularly vast, favoring the biofilm
lifestyle at the expense of the planktonic one.
• Within porous media, extensive heterogeneities of the physico-chemical properties develop. Pressure, nutrients concentrations, biocides concentrations, pH,
electric charge, rugosity of surfaces vary extensively from one pore to another,
providing a nearly infinite number of different environmental conditions. This
does not favor the development of microorganisms in one specific area, but
rather, provides ecological niches favorable to, virtually, every single species.
For example, aerobic organisms are known to live nearby the surface, where
concentrations of oxygen are relatively elevated whereas denytrifying bacteria
can live deeper in the sediments.
31
32
B I O FI L M S I N P O R O U S M E D I A
Historically, biofilms have been assumed to form continuous layers [277, 250, 63]. Other propositions suggest that
biofilms arrange in patchy aggregates within pore throats [264]. Rittmann (1993) [219]
emphasized that both representations can be correct, that is, the spatial distribution of attached microorganisms strongly depends on the physical, chemical and
biological properties of the medium and even on its history [251, 266]. For example, hydrodynamics, nutrients conditions, microorganism species, predation and
bioturbation are found to have a strong impact on the growth dynamics of biofilms
[171, 250, 219] within porous media. Thullner et al. (2004) [256] also show that growth
seems to happen predominantly within mixing zones for electron donor and acceptor.
As the biofilm grows, these zones are progressively filled giving birth to new favorable
areas. Hence, a strong coupling between the solute transport and the biofilm growth
exists. Following the same line, Knutson et al. (2006) [142] argue that transverse dispersion (in opposition to longitudinal dispersion) is fundamental and must be taken
into account for the determination of the biodegradation.
R E P R E S E N TAT I O N S O F B I O FI L M G R O W T H
C O N S E Q U E N C E S O F B I O FI L M S F O R M AT I O N
As always with biofilms, retroaction processes occur between the matrix and the environment. One caveat within
porous media is the multiscale aspect of the problem. Within subsurface soil or rocks,
or the riverine hyporheic zone local biofilm growth within the pore space can induce
substantial modifications to macroscopic mass and momentum transport dynamics
[63, 264, 288, 242, 232, 250]. Evidence of this type of modification has been developed by observing variation, over time, of macroscopic parameters such as hydraulic
conductivity and permeability as well as changes in porosity and dispersion, in conjunction with sampling indicating the presence of biofilm. For example, bioclogging
induces significant charge loss and consequently, to a reduction of the permeability
(readily ten times). This phenomenon has been the subject of many numerical and
experimental studies [54, 44, 139, 88, 257, 256, 142, 138, 223] because it is involved in
numerous applications such as biobarrier or biofouling.
3.2
M U LT I P L E - S C A L E A N A LY S I S O F S O L U T E T R A N S P O R T I N P O R O U S M E D I A
The first pioneering experimental and theoretical works regarding the transport of
non-reactive solutes have been undertaken in the 50s by Taylor [248], Aris [9], De
Josselin de Jong [80] and Saffman [224]. Theoretical works focused mainly on the
propagation of a pulse through a tube. Taylor and Aris showed that for relatively long
times, when molecules of solute have had the time to visit the entire tube section,
the transport can be described in terms of a one-equation advection-diffusion type
equation, that is, by a Gaussian shaped signal. The spreading of the Gaussian is
related to the diffusion coefficient in the diffusive regime but, interestingly, grows as
3.2 M U LT I P L E - S C A L E A N A LY S I S O F S O L U T E T R A N S P O R T I N P O R O U S M E D I A
the square of the Péclet number, in the convective regime. The Péclet number can be
defined by
Pe=
hvγ iγ d
Dγ
(3.1)
γ
where hvγ i refers to the surface average of the norm of the pointwise velocity field,
d is the diameter of the tube and Dγ is the diffusion coefficient of the solute within
the fluid γ − phase. The Péclet number is a dimensionless number comparing the
d2
characteristic times associated with diffusion D
and convection hv diγ . The effective
γ
γ
diffusivity, commonly termed dispersion at the homogenized scale, can be written
P e2
Deff = Dγ 1 +
192
(3.2)
The component of the effective diffusivity associated with the Péclet number arises
from fluctuations of the local velocity field. A molecule initially positioned in the
middle of the tube is submitted to larger velocities than the molecules nearby the
wall. As a consequence, the signal tends to spread more in the convective regime
than in the diffusive one. Later on, experiments focused on the determination of
the dispersive properties of laboratory columns and in particular, on the behavior of
dispersion in relation to the Péclet number. These early insights showed that diffusion
and dispersion are very different mechanisms. Diffusion is related to the Brownian
motion of the molecules within the bulk fluid, whereas dispersion is a complex
mixture of chemical and hydrodynamic processes that take the form of an effective
diffusion at the macroscale. The next step was to analyze data from experiments at
the field-scale [107]. These showed that, because of the multiscale heterogeneities, (1)
dispersion is not constant in relation to space and time and (2) that anomalous (nonFickian) dispersion can occur, i.e., long tails in breakthrough curves are observed and
the transport cannot be described in terms of a one-equation advection-diffusion
type equation.
The mathematical description of Darcy-scale biodegradation in porous media was
put forth in the 80s [247, 29, 30, 140]. Models were based on the assumption that the
transport could be described in terms of the concentration in the water-phase, and
did not take into account the biofilm as a different phase. Biofilm introduces a multiphase aspect in the transport processes, as well as additional heterogeneities and
length scales. A single concentration can be used in some very specific conditions, for
instance, in the local mass equilibrium situation. It corresponds to (1) biofilms thick
enough to be treated as continua and (2) hydrodynamic and chemical conditions for
which relatively small gradients of the pointwise pore-scale concentration appear.
Other conditions are termed local mass non-equilibrium and can arise from chemical,
physical or biological processes. For example, mass transfer limitation through [173]
the biofilm-fluid interface can lead to non-equilibrium situations, especially if the
33
34
Figure 13: Hierarchy of the main scales involved in solute (nutrient) transport within porous
media.
characteristic transport times within the fluid are shorter than within the biofilms.
Similarly, mass consumption within the biofilm matrix can create strong concentration gradients and non-equilibrium conditions of mass between both phases (if the
consumption is faster than the interfacial transfer [218, 4]). In the 90s and 2000s, a
substantial amount of mathematical developments have been proposed to describe
the transport of a solute undergoing biodegradation in porous media, the growth of
biofilms and the momentum transport. These models have received a lot of attention
in this thesis and are discussed in part IV.
Parameter
3.2 M U LT I P L E - S C A L E A N A LY S I S O F S O L U T E T R A N S P O R T I N P O R O U S M E D I A
0
Microscale
heterogeneities
REV
Macroscale
heterogeneities
Radius
Figure 14: The REV concept.
Nowadays, the physics of transport in porous media is widely recognized as extremely complex as it deals with multiscale heterogeneities [67]. In the context of
subsurface hydrology, processes within geological formations occur over multiple
orders of magnitude in both space and time. Scales involved in pollutant/nutrient
dispersion undertaking biodegradation in rivers subsurfaces colonized by microorganisms vary from sub-micrometer to hundreds of meters (see in Figure 13), involving
heterogeneities of permeability, absorption, reaction, dispersive properties. In principle, the mass and momentum transport equations that apply to a porous medium
could be solved at the sub-pore scale by computing numerical solutions with sufficient resolution over billions of pores. In the real world, however, this is impractical
(albeit some strategies have been developed to approximate these simulations). Primarily, the reasons for this are (1) our computational resources are limited and (2)
such detailed microscale solutions generally contain a substantial amount of information that is of low value to applications to the field. Information from the microscale
generally has to be filtered in order to get rid of high frequency fluctuations; that is,
perturbations in space or in time that represent deviations from some sort of averaged behavior. Once again two strategies have emerged. Models can either be derived
using empirical analyses or upscaling techniques. Various upscaling procedures have
been developed for this purpose, including both spatial and temporal averaging [274],
ensemble averaging [69] and homogenization [23]. In performing the upscaling, it is
35
36
always necessary to invoke one or more scaling laws [282] that indicate the form of
the redundancy in the microscale information. Usually, these scaling laws are specific
statements regarding the amplitude of the fluctuations relative to the mean, and some
notion of spatial and temporal stationarity in the processes of interest. For example,
in the context of the deterministic volume averaging theory, a local description in
space can be undertaken under the condition that the medium exhibits a hierarchy
of length scales, that is, representative elementary volumes (REVs) can be defined
and effective properties can be calculated locally on these REVs Figure 14.
Fast process
Slow process
Biofilm structure
Mass transport
detachment
-1
�de = k de
biomass decay
�d = k d-1
biomass growth
�gr = �-1
�re = S qS-1X -1
reaction
substrate diffusion
�dif = LF2 D
-1
convection
�con= LF u -1
viscous dissipation
�vis = LF2 �-1
Hydrodynamics
0.001
0.1
10
1000
100000
10000000
characteristic time, seconds
Figure 15: Temporal scales [270].
In addition to heterogeneities within the spatial continuum, processes associated
with biofilms in porous media have the particularity to be heterogeneous in relation
to the temporal continuum. Knowledge and understanding of the characteristic times
corresponding to each process is fundamental. Processes that have similar characteristic times need to be treated simultaneously whereas processes that have very
different characteristic times can be fully uncoupled. For example, let us consider
the example of the nutrient given in the previous section. On the Figure 15, it is clear
that mass transport phenomena and hydrodynamics need to be solved simultaneously whereas the modifications associated with the biofilm structure can be treated
separately.
4
SCOPE AND STRUCTURE OF THE THESIS
IN BRIEF: WHERE ARE WE
? WHERE ARE WE GOING ?
The predominant methodology that has been adopted, so far, for the study of
biofilm related problems in porous media (especially biodegradation) consists in (1)
macroscopic observations, say breakthroughs curves (BTCs), and (2) interpretation
of these in terms of either stochastic or mechanistic models. One issue, with the
mechanistics models, is that accurate predictions require knowledge of the spatial
distribution of the hydraulic, chemical, and biological properties. In other words, the
formulation of the Darcy-scale models necessitates at least some information on the
processes at the pore-scale and their spatial distribution. In most cases, however, the
characterization of the subsurface is limited. Hence, modelers tend to use simplified
heuristic transport equations involving effective parameters that are determined
using inverse optimization techniques. The obvious counterpart is that most of the
physics is lost. To overcome these difficulties, for field experiments, stochastic models
represent an interesting alternative (e.g., discussion in [163]). In the stochastic framework, the BTCs are interpreted as probability density functions in multi-dimensional,
heterogeneous domains [52, 53, 222].
However, advances in imaging techniques and modeling are allowing novel mechanistic strategies to emerge. An example of one such strategy, in the case of bioclogging,
is suggested in [18]: “(1) focus efforts on the bioclogging mechanisms that are not well
described yet in current mathematical models, (2) find the parameters and equations
that describe these mechanisms at relevant scales, (3) incorporate these components
in efficient pore-scale models based, e.g., on Lattice–Boltzmann formulations (e.g.,
[116] ), and (4) upscale the resulting models to macroscopic scales, suitable for largescale simulations”. Although this methodology has been devised to study bioclogging,
it is based on ideas that are much more general. We would rather propose the following approach (termed MVMV, for Micro-Validate-Macro-Validate, in the remainder of
the thesis), that can be used to study a broad range of biofilms related problems in
porous media, including mass transport and biodegradation:
1. image pore-scale biofilms growth within porous structures and formulate equations that describe the various pore-scale phenomena.
37
38
2. introduce these components in efficient pore-scale models based, for example,
on Lattice–Boltzmann formulations (e.g., [116]) and validate the proposed
mathematical descriptions.
3. upscale the set of differential equations (from the pore-scale to the Darcyscale) in order to obtain different macroscale models and calculate effective
parameters on the basis of 3-D realistic geometries.
4. develop the domains of validity of the various upscaled models, and validate
the theoretical analysis against Darcy-scale experiments.
Steps 1 and 2 require a direct observation of the processes at the cell-scale and at
the pore/biofilm-scale. This can be done using, for example, microfluidic devices in
conjunction with microscopy techniques that are well known. However, the ability
to directly observe biofilms within real porous structures would considerably limit
experimental artefacts, that arise, for example, from dimensionality problems (2-D vs
3-D).
Steps 3 and 4 need some specific developments, that represent the core of this
work. Darcy-scale models that are used to describe mass transport in such systems
are not fully understood yet. In addition, effective parameters can be calculated in
realistic situations, only if the spatial distribution of the different phases is known,
that is, we would need to directly image biofilms within porous media.
Our goal, in this thesis, is to develop the elements that have been missing in order
to implement such strategies. We address the development of the three-dimensional
imaging technique and of the macrotransport theory. We study both the reactive
and non-reactive situations. The reactive case obviously refers to biodegradation of
chemicals or nutrients within porous media. The non-reactive problem refers to:
1. real situations, in which the pollutant is not degraded by the microorganisms.
2. a model situation, for which the physics of upscaling is easier to understand.
The macroscopic models are validated against Darcy-scale “numerical” experiments
(direct numerical simulations at the pore-scale), under the assumption that steps 1
and 2 have already been performed, that is, we assume that we know the boundaryvalue-problem describing the mass transport problem at the pore-scale. We also
illustrate the calculation of the effective parameters on real complex geometries.
STRUCTURE
The remainder of this thesis is organized as follows.
• First, we develop a technique for imaging biofilms in opaque porous media in
three-dimensional configurations. The method uses X-ray microtomography
to delineate between the biofilm-phase, the water-phase and the solid-phase.
Part II is dedicated to the presentation of the method and discussions regarding
its scope of applications as well as its limitations. In particular, it is emphasized
that direct observations of biofilms within porous media (1) would end some
ancient debates regarding the spatial distribution of the microorganisms, (2)
provides a tool for the validation of cellular automata models in porous media
[116] and (3) captures a substantial amount of information regarding the porescale topology of the processes on a REV that can be used to develop mass
transport models on a larger scale.
• Secondly, we develop Darcy-scale mass transport models that take the spatial
distribution as an input parameters and that use this information to calculate
effective properties that arise in the macroscale balanced equations. Following
this line, the part III deals with the development of non-reactive Darcy-scale
mass transport of a solute while part IV focuses on the biodegradation (reactive) transport processes on a similar scale. In both cases, the mathematical
procedures are based on the deterministic upscaling technique termed volume
averaging with closure. The models are validated against pore-scale direct numerical simulations and the physical as well as the mathematical significance
of these models is thoroughly discussed.
• At last, in part V, we provide conclusive remarks and also discuss ongoing work
and perspectives. In particular, we present the first numerical calculations of
effective parameters (permeability) on the basis of the microtomography data.
W H AT ’ S N E W A N D W H AT ’ S N O T
?
• The methodology (MVMV), presented in this part, to study and model biofilms
related problems in porous media is new.
• The technique, presented in part II, for three-dimensional imaging of biofilms
within porous media is new.
• The non-reactive models, presented in part III, are not new BUT the theoretical
and numerical analyses that are used to determine their domains of validity
and their relationships to one another are new. We present a comprehensive
framework that did not exist and some insights to understand the physics of
the upscaling process.
• The model, developed in part IV, is new. The idea to use such a perturbation
decomposition is not new but it was never applied to any reactive case.
• The calculations of the permeabilities on the basis of realistic pore-scale geometries, as presented in part V, are new.
39
Part II
THREE-DIMENSIONAL IMAGING OF BIOFILMS IN
P O R O U S M E D I A U S I N G X - R AY C O M P U T E D
MICROTOMOGRAPHY
1
I N T R O D U C T I O N - I M P O R TA N C E O F D I R E C T O B S E R VAT I O N S
Imaging biofilms within porous media represents an important challenge. The development of microscopy methods to directly observe the microorganisms at different
scales is a sine qua non condition for understanding the mechanisms at play. The ability to image biofilm in three dimensions within porous media would also considerably
aid in
1. providing the experimental data that has been lacking in order to validate
the theoretical models that have been presented so far. For example, threedimensional imaging of biofilms in porous media would provide the information that is necessary in order to delineate the domains of validity of both
cellular automata and individual based approaches for the modeling of biofilms
structures [153] and might trigger the development of new mathematical formulations.
2. developing new models and innovative modeling strategies. Two examples of
such strategies (MVMV and Baveye’s) are given and discussed in part I.
Various methods have been developed for imaging biofilms, including confocal
laser scanning microscopy (CLSM) [155, 148] , light microscopy [14, 13], electron
microscopy [203], atomic force microscopy [21], nuclear magnetic resonance imaging
[161, 201] , infrared spectroscopy [184], optical coherence tomography [290], and high
frequency ultrasound [234]. Unfortunately, many of the aforementioned techniques
(1) are not applicable to porous media systems due to the inherent opacity of porous
structures and (2) are also not well suited for imaging regions larger than several
porous media grains. To circumvent problems (1) and (2), most of the work on porescale/biofilm-scale observations in porous media has focused on one-dimensional
or two-dimensional networks [148, 254]. There has been some discussion of the
differences induced by experimental dimensionality [254, 18, 255]. Baveye (2010)
suggests that future work should focus on three-dimensional observations, using
for example X-ray tomography, rather than on adapting pseudo one-dimensional or
two-dimensional results to three-dimensional configurations.
As a noticeable exception, Seymour et al. [229, 231, 230] used non-invasive magnetic resonance microscopy to directly observe the three-dimensional velocity field
43
44
INTRODUCTION
- I M P O R TA N C E O F D I R E C T O B S E R VAT I O N S
at the pore-scale and show that biofilm growth can induce anomalous transport. The
issue with this technique is that it does not allow spatial resolution of the pore-scale
geometry of the different phases within the porous matrix. Recent work presented
by [131] focuses on the imaging of biofilm within porous media using monochromatic synchrotron based X-ray computed microtomography. Results from this work
illustrate the ability of computed microtomography to provide experimental data
for the validation of mathematical models of porous media associated with biofilm
growth. However, the method is based so far on a cumbersome physical straining or
on attachment of a contrast agent to the biofilm surface.
Here, we present a method for imaging non a priori labeled microbial biofilms in
porous media using a benchtop X-ray computed tomography setup. The presented
method allows for the three-dimensional reconstruction of the solid, aqueous and
biofilm phases within a porous matrix with a voxel size of 9 µm (limitation only of the
scanner at hand). A significant challenge, inherent to imaging biofilm within porous
media using X-ray absorption tomography, lies in selecting proper contrast agents
to aid in differentiating between materials with similar absorption coefficients, such
as biofilm and water. Most conventional X-ray contrast agents diffuse readily into
both the aqueous phase and biofilm [131]. The proposed method focuses on the
use of a mixture of two different contrast agents that allow for differentiation of the
solid, aqueous phase and biofilm regions within the experimental systems evaluated
in this study. Herein, we will present only the final experimental procedure, that is,
material and methods that proved the most successful. However, various preliminary
tests were performed, including screening of various radiocontrast agents and of the
corresponding concentrations.
The remainder of this part is organized as follows. First, we present the different
protocols that are used in this experimental study. Then, we validate the use of the
contrast agents by comparison of two-dimensional images obtained by (1) optical
shadowscopy and (2) X-ray absorption radiography. Finally, the technique is applied
to two different model porous media experimental systems containing polyamide or
expanded polystyrene beads. Various reconstructed images are shown to illustrate
the effectiveness of the method. The limitations of the technique are discussed as
well as suggestions for future work.
2
M AT E R I A L A N D M E T H O D S
Amended river water (Garonne)
Nutrients
Air
Pump
Pump
Valves
(
(2)
Beads
~40 mm
~3.5 mm
3-D beads packings
into a plastic column
Model
Porous Medium
Part going inside the
tomograph during the
X-ray acquisition
(1)
Water tank
Compressed polystyrene beads
0.4 mm
Red square mark
behind the cell
O-ring
3 mm
PMMA plate
2-D pore network
2-D pore network
schematic view
Figure 16: Flow system used for growing biofilm within (1) the two-dimensional pore network
model and (2) the three-dimensional packed bead columns.
2.1
THE POROUS MODELS
Three types of porous media models were used for experimentation. Two-dimensional
biofilm growth experiments were conducted using a porous medium network consisting of expanded polystyrene beads (500 to 1500 µm) compressed between two
PMMA (Plexiglas®), 3 mm thick, transparent plates. Initial three-dimensional imaging was conducted using a polystyrene column (3.5 mm inner diameter) packed
with 3 mm diameter polyamide beads. Additional three-dimensional biofilm imaging
experiments were conducted using a polystyrene column (3.5 mm ID) packed with
polystyrene beads (500 to 1500 µm). Expanded polystyrene has a lower X-ray absorption coefficient than polyamide, allowing an initial contrast between the biofilm and
the beads. Schematics of the experimental devices can be found in Figure 16 and
45
46
Figure 17 for both the two-dimensional pore network and the three-dimensional
column experiments.
Camera
Fibre optic cable
Backlight
monitor
2-D pore network
Output
Input
Backlight
Figure 17: Optical visualization system for the two-dimensional pore network.
2.2
G R O W I N G B I O FI L M S
Raw water from the river Garonne (France) was collected, filtered using a 500 µm
screen, clarified via sedimentation for approximately 24 hours. The river water was
further amended with sodium acetate CH3 COONa; 3H2 O (carbon source) and
potassium nitrate KNO3 (electron acceptor) as indicated in Table 4. The prepared
water was then placed in a 200 mL plastic feed tank used as the reservoir for experimentation. The feed tank was refilled with prepared river water daily for the duration
of the experiments and constantly aerated using an air pump. The microbial flora
naturally present in the prepared river water was experimentally determined to form
sufficient biofilm with the porous medium for the purposes of this study. Flow within
the experimental systems was induced using either a peristaltic or diaphragm pump
(as detailed in Table 4). All experiments were conducted at 20◦ C ± 1◦ C in the absence
of light in order to control the growth of phototrophic organisms. Additional details
are provided in Table 4.
2.3 C O N T R A S T A G E N T
2-D pore network
3-D polyamide beads
3-D polystyrene beads
500 to 1500 µm
3mm
500 to 1500 µm
Pump
Prominent Gamma/L
Ismatec Mini-S 820
Watson Marlow 505 Du
Type
diaphragm
peristaltic
peristaltic
Introduced at days
0, 3, 6 and 9
0, 4 and 7
0, 3, 6 and 9
CH3 COONa, 3H2 O
0.66 g
0.16 g
0.66 g
KNO3
0.33 g
0.06 g
0.33 g
CBaSO4
0.33 g/mL
0.66 g/mL
0.33 g/mL
CKI
0.1 g/mL
0 g/mL
0.1 g/mL
Flow rates
3.5 mL/min
6 mL/min
0.07 and 0.5 mL/s
Beads
diameters
Nutrients
Contrast agents
Table 4: Experimental parameters.
2.3
CONTRAST AGENT
As previously mentioned, both the biofilm and the aqueous phase have similar X-ray
absorption properties. In addition, all the experimental systems evaluated in this
study were designed using plastic materials in order to minimize the total X-ray
exposure time of the microorganisms as well as to optimize the grey level scaling.
Unfortunately, the plastic beads used as the experimental porous medium also have
similar X-ray absorption properties to the biofilm and aqueous phase. Hence, obtaining contrast between the different phases requires the utilization of multiple
contrast agents. Conventional contrast agents (e.g. potassium iodide) diffuse readily
into biofilm when present in the aqueous phase. In this study, we use a medical
suspension of micrometer sized barium sulfate (Micropaque, Guerbet, see in Figure
18), conceived to have a high density and low viscosity [62, 200], to enhance the
absorption of the water-phase.
Although barium is usually highly toxic, it is commonly used as a medical radiocontrast agent for X-ray imaging of the gastrointestinal tract or angiography because of its
insolubility in water and because it is known not to diffuse within the tissues. The idea
behind the utilization of such a suspension is that particles are size excluded from
the EPS matrix. If not totally immobilized, micrometer sized cells within biofilms are
known to be greatly constrained in their motion. Hence, diffusion of similar sized
barium sulfate particles through the polymeric matrix itself is likely to be negligible.
To what extent the contrast agent can penetrate into the biofilm following the flow
within nutrients channels and how this depends on the matrix architecture remains
47
48
Figure 18: Scanning electron micrograph of the Micropaque barium sulfate particles. Courtesy
of A. Phillips, J. Connolly and R. Guerlach (Montana state university).
to be fully characterized. It is interesting to emphasize that most studies concerned
with convective flows within biofilms involve sub-micrometer sized particles [241]
and that microbes grown with micrometer sized latex beads seem to be immobilized
[87]. In addition, potassium iodide was added to the barium sulfate suspension in
order to provide the required contrast between the polystyrene beads and the biofilm.
Various ionic or non-ionic iodinated radiocontrast agents are used for medical purposes [10]. In our case, we only require that it readily diffuses within the polymeric
matrix. In this context, iodide (whether NaI or KI) has proved to be adapted to X-ray
microtomography for non-invasively imaging biological specimens [45].
For the polyamide beads, experiments focused on obtaining an important contrast
between the biofilm and the water-phases, using only barium sulfate at higher concentrations. The details of the contrast agent mixtures used during experimentation
are provided in Table 1. Preliminary scanning of the concentrations ratios were performed; herein, only the concentrations that proved to be the most successful are
presented.
It is important to keep in mind that various strategies were formerly imagined, but
only this one proved to be successful. In particular, a previous approach consisted in
using barium sulfate to enhance the X-ray absorption of the biofilm-phase (in opposition to the water-phase). The barium sulfate was present within the water during the
growth of the biofilm, and clean water was flushed in the column for the imaging. The
idea was to trap the particles within the biofilm matrix, in order to provide a contrast
between the biofilm-barium association and the water. Unfortunately, results showed
2.4 I M A G I N G P R O T O C O L S
that it was pretty hard to delineate between the portion of the barium sulfate that was
simply aggregating and the biofilm.
2.4
IMAGING PROTOCOLS
2.4.1 Two-dimensional imaging
The continuous flow of amended river water through the two-dimensional flow cells
was induced and biofilm was allowed to develop for 10 days at which point optical
imaging commenced using the system presented in Figure 17. A white LED backlight
(PHLOX®) applied a uniform illumination of the pore network from beneath the
stage and images were captured from above using a 12 bit (SensiCam) camera linked
to a computer by a fiber optic cable (as illustrated in Fig (17)). Following optical
imaging, 10 mL of the contrast agent solution consisting of 0.33 g/mL barium sulfate
and 0.1 g/mL potassium iodide was injected into the flow cell. The system was then
set to rest for approximately 1.5 hours in order to simulate the three-dimensional
X-ray tomography image acquisition time frame (see Section 2.3.2). After this delay,
a two-dimensional X-ray absorption radiograph was captured using a Skyscan 1174
tomograph with a pixel size of 12 µm.
2.4.2 Three-dimensional imaging
After 10 days of continuous flow the experimental flow cell was removed from the
water flow circuit. 10 mL of the contrast mixture, containing a suspension of barium
sulfate, potassium iodide, and water, was slowly injected through the porous model
using a syringe. The concentrations of the contrast agent additives for the various
experiments evaluated in this study are detailed in Table 1. The experimental flow
cells then sat stagnant for approximately 15 minutes in order to allow for diffusion of
the iodide into the biofilm. During these 15 minutes, a Skyscan 1174 tomograph was
set to a tension voltage of approximately 50 kV and a current of 800 µA. All computed
tomography imaging for this study was conducted at a resolution of 9 µm per pixel on
a 360◦ rotation with a rotation step ranging from 0.5◦ to 0.7◦ . In each case, the total
duration for tomographic imaging is approximately 1.5 hours. The major technical
limitation we encountered during tomographic imaging was ring artifacts, regardless
of the use of the ring artifact reduction option in the commercial software NRecon
(SkyScan). Meanwhile, there is no limitation in the method itself which prevents the
utilization of synchrotron based tomography (monochromatic) or new generations
of scanners capable of producing higher quality images.
49
50
2.5
D ATA A N A LY S I S
2.5.1 Two-dimensional image analysis
Two-dimensional (radiographic) X-ray absorption images (12 bit TIFF images) and
two-dimensional optical images (12 bit TIFF images) were post-processed using
the open source software package ImageJ. For the X-ray images, we applied a FFT
(Fast-Fourier Transform) bandpass filter to reduce extreme frequency noise. Then,
the two data sets are compared using pseudocoloration based on a LookUp Table
(LUT). This coloration was chosen on the basis of visualization purposes, as guides
for the eyes. Quantitative measures, such as correlation ratios, strongly depend on
the segmentation procedure. This is beyond the scope of this work to propose such
methods; rather, we provide a qualitative analysis of the results. Representative images
used for comparison of the two data sets are provided in Figure 19.
2.5.2 Three-dimensional tomography
The absorption projection images (12 bit TIFF images) were reconstructed using
NRecon to obtain a set of cross-sectional slices (16 bit TIFF images) of the columns,
using ring artifact and beam hardening correction. The various greyscale images
presented in this article are encoded as 8 bit images for visualization purposes. For
the polyamide beads, images were slightly smoothed and undergo global binarization
using ImageJ Otsu’s method. The surfaces are built in p3g, surface format, using
the commercial software CTAn (Skyscan) and the three-dimensional geometry is
observed using the software CTVol (Skyscan). The goal of this work is to provide
an operational technique for imaging biofilm, i.e., to demonstrate that the use of a
barium sulfate suspension as a contrast agent is feasible for imaging biofilm within a
porous medium matrix.
3
R E S U LT S
3.1
TWO-DIMENSIONAL EXPERIMENTS
(1) Optical
(2) Radiograph
Figure 19: Comparison of two-dimensional images after 10 days of growth obtained using (1)
the visualization device detailed in Figure 17 and (2) Skyscan 1174 X-ray absorption
radiograph captured approximately 1.5 hours after injection of the contrast agent
mixture. Three zones A, B and C, assessing various pore-scale geometries, have
undertaken pseudocoloration using ImageJ on the basis of a LookUp Table (LUT).
The dark-blue parts correspond to the beads, the blue-green-brown parts to the
biofilm and the white parts to the aqueous phase. The parts circled in red on the
radiograph correspond to whether detached pieces of biofilm or gaz bubbles that
are not present on the optical shadowscopy.
The purpose of the two-dimensional investigation was to evaluate the behavior of
the contrast agent mixtures and to ensure that sufficient contrast between the various
phases was achieved.
Potential issues identified include:
51
52
R E S U LT S
POTENTIAL ISSUE A
Exclusion of the barium sulfate suspension from the biofilm
EPS needs to be verified.
The contrast agents need to be investigated to see whether
interactions between the microorganisms and the contrast agents modify the EPS
geometry, thereby preventing the acquisition of representative images.
POTENTIAL ISSUE B
POTENTIAL ISSUE C
The injection of the contrast agent mixture needs to be
examined to determine whether the induced shear stress associated with injection
results in biofilm detachment from the porous media matrix or in modifications of
the EPS geometry.
POTENTIAL ISSUE D
It is necessary to determine whether prolonged (1.5 hour)
X-ray exposure induces changes in the EPS geometry.
One caveat that must be taken into account when considering the presented X-ray
computed tomography imaging method for biofilm investigations is that the technique is non-invasive, in that the biofilm growth can be imaged in situ, however X-ray
exposure is expected to either severely retard microbial growth or kill the microorganisms all together. Thus the technique can be considered non-destructive to the
porous media-biofilm matrix, however the imaging technique is still terminal. In
order to investigate potential temporal changes to the biofilm matrix during imaging, a series of experiments were conducted to assess whether the issues previously
identified as Problems A, B, C, and D negatively impact image accuracy and quality
on the time-scale of a three-dimensional tomography acquisition (approximately
1.5 hours of exposure time using the Skyscan 1174 tomograph). Thus, images of a
two-dimensional pore network colonized by biofilm obtained using both optical
shadowscopy and X-ray computed tomography were compared. Results of the twodimensional investigation are presented in Figure 19. Three zones, corresponding to
different biofilm geometries, have been processed using a pseudocoloration to allow
for comparison. Within Zone A, three biofilm filaments are clearly visible on both
the optical image as well as the X-ray image. In Zones B and C, a clear correlation
between the two geometries is apparent although discrepancies between the optical
image and X-ray tomography image exist within these zones as well. Based upon the
qualitative image comparison within these zones there appears to be good agreement between the two image capturing methods. Since the optical imaging method
focuses, primarily, on a top-side view of the biofilm, the increased distribution of
barium sulfate within the radiograph can be attributed to an increased flow channel
volume within the biofilm that is not visible within the depth of field captured using
optical microscopy. Thus the qualitative results presented in Figure 19 illustrate the
utility of using X-rays (and the chosen contrast agents) to image biofilm, particularly
when three-dimensional tomographs are captured as opposed to two-dimensional
radiographs since the tomographs are capable of providing direct visualization of
3.2 R E S U LT S O F T H E 3 - D T O M O G R A P H Y A N D D I S C U S S I O N
the channeling suspected to be present within the biofilm present in Zones A, B, and
C. The barium sulfate suspension used for imaging does not appear to significantly
enter the EPS layer within these zones. Rather the barium sulfate appears to follow
the aqueous phase flow channels. These conclusions are supported by the results,
provided in the next section, concerning the successive use of barium sulfate and
iodide. Hence, the issues previously detailed as A and B do not appear to significantly
affect our imaging results. However, further investigations are required in order to
elucidate the microscale behavior of the particles, especially in relation to the density of the EPS matrix and the physical properties of the contrast agent suspension.
Nevertheless, the use of barium sulfate as a contrast agent for imaging biofilm within
porous media is promising since the delineation of the topology of the flow channels
and the associated impact on the transport processes at the pore-scale is definable
within relatively large volumes.
While Problem D cannot be fully addressed using this two-dimensional experiment,
we observed no substantial modifications to the EPS geometry after approximately
30 minutes of X-ray exposure. While biofilm associated microorganisms are expected
to be severely inhibited or killed by exposure to X-rays, the biofilm matrix appears
to be stable after exposure times of up to 1.5 hours from the benchtop tomography
(Skyscan 1174) X-ray source used in this investigation. Three-dimensional results
concerning this aspect of the problem are discussed in the next section.
3.2
R E S U LT S O F T H E
3-D
TOMOGRAPHY AND DISCUSSION
3.2.1 Single polyamide bead
The first set of 3-D experiments focuses on imaging of biofilm on 3 mm diameter
polyamide beads. For this case, only the barium sulfate suspension was introduced
as a contrast agent. Examples of projection data are presented in Figure 20 at time
t = 0 without biofilm and at t = 10 days following the biofilm growth phase. Differences between these two raw images take the form of patchy white spots meaning,
locally, lower X-ray absorption. These zones appear because biofilm has developed,
constraining the local volume available for barium sulfate. This set of absorption
data is used to reconstruct a set of cross-sectional slices on a single bead within the
experimental column. Greyscale images as well as representative binary images are
provided in Figure 21 at t = 0 and Figure 21 for t = 10 days. At t = 0, a cross-sectional
circular shape, corresponding to the polyamide bead, is observed. After 10 days of
biofilm growth, the boundary of the object that we imaged is tortuous and covers
more surface. On the basis of the two-dimensional study presented in the preceding
section, we interpret this additional area as biofilm. It is important to note that within
Figure 21 there is no contrast between the plastic bead and the biofilm grown on
the bead, further reinforcing the proposition that the barium sulfate suspension is
excluded from the EPS layer of the biofilm. A solution of potassium iodide was then
53
54
R E S U LT S
t=0 days
t=10 days
Figure 20: Examples of projection images obtained with the SkyScan 1174 using BaSO4 as the
contrast agent at time t = 0 days on the left and t = 10 days on the right. Both images
have undertaken a pseudocoloration in ImageJ on the basis of the unionjack LUT
(only for visualization purposes). Blue corresponds to the highest X-ray absorption,
red to intermediate absorption and white to lowest absorption.
flushed through the system. A depiction of the polyamide bead after potassium iodide
addition is provided as Figure 21. Iodide, when present in the aqueous phase, diffuses
readily into biofilm present within the pore space. As a result, the contour of the
polyamide bead is all that is visible in Figure 21, thereby confirming that the tortuous
zone surrounding the bead in Figure 21 is in fact biofilm.
Surface reconstructions of the polyamide bead are provided in Figure 22. The
surface reconstructions correspond to t = 0, prior to biofilm growth, and t = 10 days,
after the biofilm growth phase. Contrast for both images is provided using the barium
sulfate suspension. Within the imaged section, the biofilm appears to be highly
heterogeneous and represents about 6% of the volume of the naked polyamide bead.
Additional study is required in order to draw further conclusions on biofilm growth
and development within our experimental system, however, the ability to image
biofilm within porous media using the proposed technique has been established,
which is the purpose of this study.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 21: Cross-sectional reconstructed X-ray computed tomography data for a polyamide
bead at t = 0 days with BaSO4 (a) and (b); for a polyamide bead at t = 10 days, after
biofilm growth, with BaSO4 as the contrast agent (c) and (d); for a polyamide bead
at t = 10 days using potassium iodide as the contrast agent (e) and (f). Images (a), (c)
and (e) are grey scale initial images and (b), (d) and (e) their binarized counterpart
(Otsu’s algorithm with ImageJ).
3.2.2 Results for the polydisperse expanded polystyrene beads
For more complex porous structures, such as polydisperse polystyrene beads, the
alignment of tomography data captured both prior to, as well as following biofilm
55
56
R E S U LT S
t=0 days
t=10 days
Figure 22: Three-dimensional surface reconstructions of the polyamide bead at time t =
0 using BaSO4 as the contrast agent and the biofilm (soft blue-green) and the
polyamide bead (dark) at time t = 10 days.
t=0 days
t=10 days
Figure 23: Greyscale cross-sectional X-ray computed tomography for the experimental
columns packed with polystyrene beads at t = 0 with the mixture of contrast agents
and at time t = 10 days.
growth is not necessarily possible due to the potential for bead displacement due to
fluid transport or biofilm growth. Thus image processing techniques such as image
subtraction are not applicable. Image subtraction is not also desirable when working with natural media with heterogeneous solid phase X-ray absorption properties.
Therefore, we developed a more direct technique. A mixture of the barium sulfate
and potassium iodide contrast agents at two different concentrations was utilized to
differentiate between the three materials present within the experimental system. Using this contrast mixture, tomographic imaging was performed. Preliminary imaging
was carried out at time t = 0 after introducing the contrast agents mixture. Imaging
was also conducted at time t = 10 days, approximately 15 minutes after injecting the
mixture of both contrast agents. Comparative results are provided in Figure 23 for
the two data sets. Results for the t=0 data set indicate that the contrast agent solution
delineates, clearly, the beads contained within the column. At t= 10 days the presence
of three distinct phases is observed. The brightest phase corresponds to the barium
sulfate (highest absorption coefficient). The dark regions correspond to beads and the
intermediate greyscale values are interpreted as biofilm which the iodide has diffused
into. Figure 24 illustrates the results of a comparative experiment examining biofilm
growth within packed bead columns through which two different flow rates were
applied. For this experiment two columns containing polystyrene beads and connected to the same water supply were exposed to flow rates of 0.07 mL/s and 0.5 mL/s.
Within the two columns, biofilm growth is seen to decrease with increasing flow rate.
While additional experiments are required in order to draw conclusions about biofilm
growth within porous media, the presented results demonstrate that pore-scale information on biofilm growth within a porous medium is readily achievable using the
proposed imaging method. Using the results generated using the presented method
calculations of column or regional permeability can be performed numerically by
solving Navier-Stokes equations. Darcy-scale dispersion tensors can also potentially
be calculated using upscaling techniques.
Successive imaging of a single column was conducted in an effort to further evaluate the effect of X-ray exposure on biofilm structure (Problem D). The total exposure
time was 3 hours and consisted of a sequence of 2 imaging cycles. At the conclusion
of this experiment no change within the biofilm geometry were observable. This suggests that, for an acquisition time greater than 1.5 hours, X-rays at the energy emitted
by the Skyscan 1174 tomograph (50 kV and 800 µA) do not modify the geometry of
the biofilm EPS matrix.
57
(a) Q=0.07 mL/s
(b) Q=0.5 mL/s
Figure 24: Examples of reconstructed (X-ray Skyscan 1174 data) sectional slices for the entire
length of the column obtained after 10 days at a flow rate of approximately (a)
Q=0.07 mL/s and (b) Q=0.5 mL/s (a pseudocoloration has been applied to the
images using ImageJ on the basis of the ceretec LUT and only for visualization
purposes). The white-red parts correspond to the beads, the red parts to the biofilm
and the blue-dark parts to the aqueous phase.
58
4
CONCLUSION AND DISCUSSION
In this study, we present first results for a new method for imaging
biofilm in porous media using X-ray computed tomography. We successfully used
a mixture of two different contrast agents to obtain a three-phase contrasted threedimensional representation of a model porous medium containing solids, water and
biofilm. This approach, because of its simplicity, accessibility and applicability to
complex porous structures, provides an interesting and versatile framework for studying biofilm within porous media systems. The method can potentially be used in the
calculation of porous media effective parameters. In particular, the presented method
opens possibilities for systematic studies of biofilm response, within porous media,
to changes in physical, chemical and biological parameters. For example, modifications of local Reynolds and Péclet numbers, nutrient availability, temperature and pH
stresses, and the impact of biofilm biodiversity on biofilm geometry within the threedimensional porous media matrix can potentially be investigated. While the use of
synchrotron X-ray sources hold the potential to provide higher quality imaging data
and the imaging of biofilm in porous media has been investigated using synchrotron
light and silver microspheres as a contrast agent [131], the method presented in this
study is functional using both benchtop tomographs, such as the Skyscan 1174 as
well as synchrotron X-ray sources, even though more sophisticated image processing
procedures need to be developed. Thus the presented method is broadly applicable
since imaging is not necessarily restricted by synchrotron accessibility and beam time
constraints.
On the other hand, one significant limitation associated with the use of benchtop
tomographs is that the required imaging time for porous media materials such as
glass beads, soil or rock materials is significantly greater than the 1.5 hour image
acquisition time reported in this study. As a result, investigations using these types of
porous media are anticipated to require synchrotron light sources.
Future work will focus on (1) optimization of the image acquisition techniques such
that images that can be easily (and impartially) segmented into their respective phases
are obtained (whether it is using a different polychromatic or a monochromatic
imaging system, optimizing the concentrations of the contrast agents, using separate
imaging of the solid phase, etc.) (2) a comparison of this work with other threedimensional planar imaging techniques such as confocal laser scanning microscopy,
CONCLUSION
59
60
CONCLUSION AND DISCUSSION
for instance to provide further understanding of the interaction between the 1 µm
BaSO4 suspension and the architecture of the biofilm, (3) application to real porous
samples with heterogeneities of absorption coefficients in the porous structures, and
(4) an investigation of microbial retardation or mortality induced by X-ray exposure.
? In this thesis, we are particularly interested in using this information to model mass transport within porous media with
biofilms. How can we use the 3-D images obtained using X-ray microtomography for
this purpose ?
First, we need to define exactly what are the processes that we want to describe
at the pore-scale level, that is, we need to derive a boundary value problem describing the mass balance within the system. For mass transport, it usually can include
multiphasic descriptions of processes such as diffusion, convection, absorption or
reaction. Generally, these involve complex non-linear mechanisms that are extremely
difficult to upscale. Hence, this microscale problem needs to be defined carefully and
often requires simplifications that must be clearly emphasized.
Once the small scale mathematical description of the problem is given, one can
average the partial differential equations that are used to model the mass transport.
Various upscaling techniques can be used toward that end, and are thoroughly discussed within the next part. These are aimed to determine the macroscale counterpart
of the microscale boundary value problem. Under the assumption that a local (in
space) description is desired, i.e., that a representative elementary volume can be
defined, the effective parameters (for example dispersion or permeability) of the
macroscale models can be calculated on the basis of this REV. In our case, we use the
X-ray tomography technique in order to obtain the pore-scale topology of the processes on a REV, and use this as an input parameter for upscaled models. Deriving a
macroscale theory that applies to solute transport is the goal of parts III (non-reactive)
and IV (biologically mediated).
H O W C A N W E U S E T H I S I N F O R M AT I O N
Part III
MODELING NON-REACTIVE NON-EQUILIBRIUM
MASS TRANSPORT IN POROUS MEDIA
1
INTRODUCTION
In this part, we are interested in the Darcy-scale description of the transport of a
conservative (non-reactive) solute tracer in porous media with biofilms. As previously
emphasized, this case can refer to a physical situation in which the chemical species
are not biodegraded. However, this is not the primary interest of this part. Here, we
aim to understand the theory of volume averaging, and the different models that can
be derived using this theory. The non-reactive problem is used as a simple case, in
order to understand the upscaling process itself. We will focus on a biofilm-specific
reactive situation in the next part, in which we use our knowledge of the upscaling
process to develop a one-equation model that lends itself very well for the biofilm
problem.
Following the same line, we also adopt a general formalism in this part, rather than
a biofilm-specific one. For example, upscaling mass balanced equations from the
pore-scale to the Darcy-scale in a dual-phase porous medium is, mathematically
speaking, equivalent to upscaling from the Darcy-scale to a larger scale in a dualregion porous medium (Fig. 25). Hence, we consider a general framework defined by
the mathematical structure of the boundary value problem at the small scale, rather
than by the scale itself and the corresponding physical phenomena. Our goal is to
develop a macroscale solute transport (via convection and diffusion) theory for discretely hierarchical and highly heterogeneous porous media containing two different
regions/phases (Fig. 25). Depending on the value of the ratio of the characteristic
times associated to the processes occurring in each region/phase and on the scale
of application, these systems are usually referred to as mobile-immobile, mobilemobile, dual-porosity, dual-permeability or, in a more general way, dual-continua
[7, 43, 47, 48].
This part is organized as follows:
1. In this section, we briefly introduce the mathematical problem, discuss the
various models and present the scientific scope for this part.
2. In section 2, we develop our macroscale transport theory and discuss the limitations associated to each model on the basis of theoretical analyses.
63
Figure 25: Hierarchy of the different scales for dual-region large-scale averaging (on the left)
and dual-phase Darcy-scale averaging (on the right)
64
1.1 M I C R O S C O P I C D E S C R I P T I O N O F T H E P R O B L E M
3. We advance a formal proof, section 3, for the equivalence between volume
averaging and moments matching techniques regarding one-equation nonequilibrium models.
4. In section 4, we compare numerically on a simple two-dimensional example,
the behavior of the models in response to a square input signal and delineate,
on the basis of these computations and of the theoretical analysis carried out
in the previous section, their domains of validity.
1.1
MICROSCOPIC DESCRIPTION OF THE PROBLEM
Herein, we present the boundary value problem that is used to describe the mass
transport at the microscale (pore-scale/biofilm-scale for the biofilm problem, Darcyscale for the larger scale problem). We consider two different phases/regions in which
the solute undertakes diffusion/dispersion and convection. On the grains surface the
flux is zero, that is, the solid σ is impermeable (this boundary condition BC4 exists
only in the case of a pore-scale description and not for larger scales, unless when there
are impervious nodules). We assume a continuity of the concentrations and of the
flux at the boundary between the phases γ and ω. We consider only a diffusive flux on
this boundary. This assumption is classically used for the biofilm transport problems,
although an advective flux can be developed in the case of channeled biofilms if the
matrix plus the channels are treated as a single continuum at the pore-scale [11].
For the sake of simplicity, we will consider that this (diffusive flux on the boundary)
approximation is valid for both the biofilm and the larger scale problem. In essence,
this means that the microscopic characteristic times associated with the advective
terms in each phase/region are separated by several orders of magnitude, resulting
essentially in a diffusive flux at the interface. In addition, we will not consider complex
boundary conditions, such as jumps or conditions on chemical potentials. Although
such boundary conditions would probably capture more physics of the problem, we
aim to keep it simple in order to study the dual-continua upscaling process itself,
more than the physics at the pore-scale.
The mass balanced equations take the form
γ − phase :
BC1 :
BC2 :
BC3 :
BC4 :
∂cγ
+ ∇ · (cγ vγ ) = ∇ · (
∂t
−nγσ ·
Dγ · ∇cγ)
Dγ · ∇cγ = 0
cω = cγ
−nγω · γ · ∇cγ = −nγω ·
−nωσ · ω · ∇cω = 0
D
D
Dω · ∇cω
(1.1)
on S γσ
(1.2a)
on S γω
(1.2b)
on S γω
(1.2c)
on S ωσ
(1.2d)
65
66
INTRODUCTION
ω − phase :
∂cω
+ ∇ · (cω vω ) = ∇ · (
∂t
Dω · ∇cω)
(1.3)
In these equations, ci is the concentration in the i − phase supposed to be zero at
t = 0. The velocity vector vi is supposed to be known pointwise for the purposes of
D
this study. i is the diffusion (pore-scale)/dispersion (Darcy-scale) tensor. S ij is the
interface between the i − phase and the j − phase and Sij is the surface; nij is the
corresponding normal vector pointing from i to j.
In the biofilm case, this representation is built on two assumptions: 1) the biofilm is
thick enough to be treated as a continuum [287, 285], and 2) the microscale term ∇ ·
(cω vω ) describes the velocity induced by the nutrients channels that sometimes form
within the biofilms. Such a description can be undertaken under the condition that
there is a separation of the length scales corresponding to the diameter of the channels
and to the width of the biofilm (the biofilm must be very large as compared to the
size of the channels). These channels can also be treated, technically, as part of the
fluid-phase in which case one can simply impose vω = 0 pointwise. The momentum
transport can be modeled, at the pore-scale, using Navier-Stokes equations. For the
purposes of this study, we will consider that the velocity field is known pointwise in
the whole system. We will also assume that we are dealing with incompressible flows,
which straightforwardly leads to an equation for the conservation of the mass that
reduces to ∇ · vi = 0.
For the dual-region problem, the microscale momentum transport equations already represent an homogenized problem that is often described in terms of a simple
Darcy’s law Equation (1.4).
vi = −
Ki
µi
(∇Pi − ρi g)
(1.4)
Other formulations are possible such as Darcy-Forchheimer’s or Brinkman’s, depending on the physical situation at the pore-scale.
∂ε
In addition, to simplify the εi from the equations, we assume ∇εi = 0 and ∂ti = 0.
The conservation of the mass can also be written ∇ · vi = 0 under the condition that
the porous medium is homogeneous, that is, again, ∇εi = 0.
1.2
F U L LY N O N - L O C A L M O D E L S
Upon averaging the microscopic boundary value problem, the macroscale transport
process exhibits time and spatial non-locality under the most general conditions
[144, 145, 68, 177, 43]. This behavior can be mathematically described via solutions
or integro-differential equations that involve convolutions of a memory (kernel) function over space and time, i.e., the local solution depends on the solution everywhere
in space and time, hence the word non-local. The non-locality is a rather complex
1.2 F U L LY N O N - L O C A L M O D E L S
Figure 26: Simple illustration of the non-locality concept.
concept and, in addition to the formal definition, it is useful to give an example that
can help in its comprehension. For instance, let us consider the propagation of a Dirac
distribution, through a porous medium as represented on Fig (26). In the first stages
of the propagation, the signal “sees” a very small portion of the porous structure that
changes over space and time. The transport properties such as dispersion, permeability, volumic fraction of the phases can be defined only in relation to the signal
itself and its history. The transport properties at one point of the medium change over
space/time and depend on the shape of the signal, which is determined by what has
happened before. In such situations, the transport is referred to as non-local. Going
back to our example, the signal will spread until, eventually, it sees a representative
portion of the porous medium. The medium that is seen by the signal will never be
exactly the same but, if there is hierarchy of length scales, as presented on Fig (26),
the transport properties will be identical in an averaged sense, and the transport
becomes local.
One such example of a non-local model is given in [144] and can be written Equation (1.5)
∂ hhcii
+ ∇ · hhqii = hhSii
∂t
(1.5)
with
q = vc −
D · ∇c
(1.6)
67
68
INTRODUCTION
where the ensemble average hh·ii refers to an average over an ensemble of velocity
fields, often interpreted, for real field applications, in terms of its volume averaged
counterpart under some ergodic conditions. If ergodicity does not apply [70, 71, 98],
other volume averaged non-local theories have been formulated, for example in [282].
In both cases, non-local transport theories necessarily involve convolutions over
space and time, and Eq (1.5) is only taken as an example. c is the pointwise microscale
concentration, v is the microscale velocity field, is the microscale diffusion tensor
and S is the microscale source term. The macroscopic flux can be written Equation
(1.7)
D
Z
hhqii = V hhcii − dx1
Zt
"
dt1
−∞
D (x − x1 , t − t1 ) · ∇ hhc (x1 , t1 )ii
(1.7)
#
∂ hhc (x1 , t1 )ii
+ V · ∇1 hhc (x1 , t1 )ii − σ (x1 , t1 ) hhS (x1 , t1 )ii
+w (x − x1 , t − t1 )
∂t1
D
in which V is a weighted averaged velocity; , w and σ are averaged quantities that
depend mostly on a Green’s function that can be calculated through the resolution
of a microscale boundary value problem. The difficulty with such fully non-local
theories is that the resulting macroscale problem Equation (1.5) exhibits, under most
circumstances, almost as many degrees of freedom as the original microscale problem.
These representations are extremely important from a fundamental perspective, but
they may not always significantly reduce the information content of the underlying
microscale model.
Other solutions to get an accurate response include direct numerical simulations,
but it is not often tractable for most of porous systems because of the complexity
involved. Degraded models, but still rather accurate if one is concerned with the
estimate of the flux exchanged between the different phases/regions, can be derived
under the form of mixed models for mobile/immobile systems, i.e., systems with high
diffusivity contrast and no advection in the low diffusivity phase/region [8, 159, 83].
Anyways, these formulations are often overly complex and, hence, it is of utmost
importance to develop local models that approximate the convolutions under some
asymptotic limitations, whenever it is possible. Toward this goal, various degrees
of approximations have been proposed for non-local models (e.g., [43, 121]), usually based on various conjectures about the time and space scales of the transport
phenomena involved. Developing these space local models and understanding their
limitations is the goal of this part. Various techniques are available to perform such
operations. Herein, we use the filtering process that consists in (1) volume averaging
over a locally periodic representative elementary volume (REV) and (2) closing the
problem by imposing a relationship between the fluctuations at the small scale and
the macroscopic variables. We also provide a theoretical analysis in terms of the
spatial moments matching technique.
1.3 V O L U M E AV E R A G E D E FI N I T I O N S F O R T H I S W O R K
1.3
V O L U M E AV E R A G E D E FI N I T I O N S F O R T H I S W O R K
To obtain a macroscopic equation for the mass transport, we average each microscopic equation over a representative region (REV), for example at the Darcy-scale, V
Fig (25). Vγ and Vω are the Euclidean spaces representing the γ- and ω-phases within
the REV. Vγ and Vω are the Lebesgue measures of Vγ and Vω , that is, the volumes of
the respective phases. The Darcy-scale superficial average of ci (where i represents γ
or ω) is defined the following way
1
hcγ i =
V
Z
1
cγ dV , hcω i =
V
Vγ
Z
Vω
cω dV
(1.8)
Then, we define intrinsic averaged quantities
1
hcγ i =
Vγ
γ
Z
1
cγ dV , hcω i =
Vω
Vγ
Z
ω
Vω
cω dV
(1.9)
Volumes Vγ and Vω are related to the volume V by
εγ =
Vγ
Vω
, εω =
V
V
(1.10)
with εγ and εω called volume fractions of the relevant phases/regions.
To simplify the developments, we assume that both volume fractions are constant
through time and space (we have already discussed and justified this quasi-steady
assumption in the case of growing biofilms). In addition, we have
hcγ i = εγ hcγ iγ , hcω i = εω hcω iω
(1.11)
During the averaging process, there arise terms involving the point values for
cγ , cω , vγ and vω . To treat these term conventionally, one defines perturbation
decompositions as follows
cγ = hcγ iγ + c̃γ
cω = hcω iω + c̃ω
(1.12)
(1.13)
γ
(1.14)
ω
(1.15)
vγ = hvγ i + ṽγ
vω = hvω i + ṽω
Under the assumption that the transport is local, we have the following properties
for the perturbations
69
70
INTRODUCTION
hc̃γ iγ = 0
hc̃ω iω = 0
In addition, we define a volume-fraction weighted averaged concentration [285,
188]
hciγω =
εγ
εω
hcγ iγ +
hcω iω
εγ + εω
εγ + εω
(1.16)
Perturbation decompositions associated with this weighted averaged concentration are
cγ =hciγω + ĉγ
cω =hci
γω
(1.17a)
+ ĉω
(1.17b)
With this latter definition, we do not generally have the condition that the intrinsic average of the deviation is zero, i.e., hc̃γ iγ , hc̃ω iω = 0. However, we do have a
generalization of this idea in the form
εω hĉω iω + εγ hĉγ iγ = 0
1.4
(1.18)
T W O - E Q U AT I O N M O D E L S
T W O - E Q U AT I O N Q U A S I - S T E A D Y M O D E L
One set of approximations of the convolutions Eq (1.5) leads to the description of the transport as a system of two equations
coupled by a linear exchange operator Eqs (1.19)-(1.20) [2, 48, 47].
εγ
∂hcγ iγ
+ εγ Vγγ (∞) · ∇hcγ iγ + εγ Vγω (∞) · ∇hcω iω
∂t
= εγ ∇ · (Dγγ (∞) · ∇hcγ iγ )
+ εγ ∇ · (Dγω (∞) · ∇hcω iω )
− h (∞) (hcγ iγ − hcω iω )
+ εγ Qγ (x, t)
εω
∂hcω iω
+ εω Vωγ (∞) · ∇hcγ iγ + εω Vωω (∞) · ∇hcω iω
∂t
(1.19)
= εω ∇ · (Dωγ (∞) · ∇hcγ iγ )
+ εω ∇ · (Dωω (∞) · ∇hcω iω )
− h (∞) (hcω iω − hcγ iγ )
+ εω Qω (x, t)
(1.20)
1.4 T W O - E Q U AT I O N M O D E L S
In these equations, the macroscale parameters Vij (∞) and Dij (∞) are effective
velocity and dispersion tensors of the two-equation model; and the parameter h (∞)
is a macroscale mass exchange coefficient. The term Qi is a source term that accounts
for particular boundary and initial conditions.
Even though the model Eqs (1.19)-(1.20) captures a lot of the characteristic times
scales (or, equivalently, the eigenvalues of the temporal portion of the problem)
associated with the transport processes, such a formulation still loses important
features. The main assumptions underlying the two-equation model are that (1) the
hypothesis of separation of length scales is valid, and (2) the time scale associated
with the macroscale concentrations, hci ii , are very large as compared to the characteristic time for the relaxation of the fluctuations, c̃i (defined by c̃i = ci − hci ii ).
In particular, it has been pointed out that the first-order constant mass exchange
coefficient is applicable under certain limitations [192]. It must be emphasized that
the macroscale mass transfer coefficient, h, depends on both the medium properties
and the boundary/initial conditions, and that this parameter is generally a dynamic
one that changes in time as the solution evolves. As an effort to account for this transient behavior, various models, aiming to filter less information from the microscale
and to capture more characteristic times associated with interphases transfers have
been developed. These models can be grouped together, and are known generally as
multi-rate mass transfer models [120].
T W O - E Q U AT I O N T R A N S I E N T C L O S U R E
One other interesting idea that deserves additional consideration is one that aims
to estimate one or the other convolution at a time. Traditionally, these two hypotheses have been used together, that is, both the convolutions in space and in time are
approximated in the same model. However, fully eliminating spatial non-local parameters while keeping the time convolutions can lead to very interesting formulations.
In particular, we are interested in the time non-local model Eqs (1.21)-(1.22) proposed in [177, 237]. In this model, the hypothesis of quasi-stationarity of the closure
problems is not required. The result is that the macroscopic equations contain time
convolutions.
εγ
∂hcγ iγ
∂
∂
+ εγ Vγγ (t) · ∗
∇hcγ iγ + εγ Vγω (t) · ∗
∇hcω iω
∂t
∂t
∂t
∂
Dγγ (t) · ∗ ∂t
∇hcγ iγ
=
εγ ∇ ·
+
εγ ∇ ·
−
h(t) ∗
∂
Dγω (t) · ∗ ∂t
∇hcω iω
+
∂
(hcγ iγ − hcω iω )
∂t
εγ Qγ (x, t)
(1.21)
71
72
INTRODUCTION
εω
∂hcω iω
∂
∂
+ εω Vωγ (t) · ∗
∇hcγ iγ + εω Vωω (t) · ∗
∇hcω iω
∂t
∂t
∂t
∂
Dωγ (t) · ∗ ∂t
∇hcγ iγ
=
εω ∇ ·
+
εω ∇ ·
∂
Dωω (t) · ∗ ∂t
∇hcω iω
−
+
∂
(hcω iω − hcγ iγ )
∂t
εω Qω (x, t)
(1.22)
h(t) ∗
Here the operator ∗ indicates the convolutions in time. When the characteristic
time for the relaxation of the effective parameters, in particular of the mass transfer
function, h(t), is very small as compared to the characteristic time for temporal
variations of hci ii , the convolutions tend toward a simple integration leading back
to Eqs (1.19)-(1.20) (cf. Section 4.1 and 4.2 in [43]). Notice that we will use a global
notation ij to indicate either ij (t) or ij (∞) when there is no ambiguity as to
which one we are referring to.
An important limitation with this approach is the postulate that the non-locality in
time is weakly coupled with the non-locality in space. Significant coupling can arise
from particular choices of boundary and initial conditions; as an example, coupling
would be expected for a delta pulse. As the signal propagates through the medium,
the mass spreads in space and in time and the influence of the non-locality is often
reduced. Hence, the convolution formulation given by Eqs (1.21)-(1.22) refers to
situations in which the perturbations are well captured by the first-order spatial
closure associated with a two-equation model but in which a fully transient closure is
necessary.
Another problem with Eqs (1.21)-(1.22) or even with Eqs (1.19)-(1.20) is that the
system of coupled equations that needs to be solved is rather complex. Discretization
of the convolutions can be performed, but a computational solution would be quite
time-consuming. As a consequence, Eqs (1.21)-(1.22) are generally avoided, albeit
they have been used to interpret some experimental data [147]. Even Eqs (1.19)-(1.20)
are often being replaced by a simple advection-dispersion equation.
D
1.5
D
D
O N E - E Q U AT I O N M O D E L S
These two-equation models showed good agreement in numerous works with both
numerical computations and experimental data [55, 81, 115, 35, 57] and can be
confidently used to explore some physical aspects of the problem. In particular, two
very different situations leading to simple Fickian dispersion have been identified
[150, 205, 48].
O N E - E Q U AT I O N L O C A L M A S S E Q U I L I B R I U M M O D E L
On the one hand, when the macroscopic concentration in one phase/region can be
expressed as a thermodynamic function of the concentration in the other phase/re-
1.5 O N E - E Q U AT I O N M O D E L S
gion, the situation is called local mass equilibrium [205] and a one-equation model
Eq (1.23) represents a reasonable approximation of the mass transport problem.
∂ hciγω
+ V · ∇ hciγω = ∇ · {
∂t
(1.23)
εγ
εω
hcγ iγ +
hcω iω
εγ + εω
εγ + εω
(1.24)
(εγ + εω )
Dequ · ∇ hciγω}
where
hciγω =
O N E - E Q U AT I O N T I M E - A S Y M P T O T I C N O N - E Q U I L I B R I U M M O D E L
On the other hand, the time-infinite comportment of the first two spatial centered
moments reveals that two-equation models have a time-asymptotic behavior which
can be described in terms of a classical advection/dispersion one-equation model Eq
(1.25) [295, 2]. It corresponds to a particular time constrained local non-equilibrium
situation for which the tracer does not exhibit anomalous dispersion.
(εγ + εω )
∂ hciγω
+ V · ∇ hciγω = ∇ · {
∂t
D∞ · ∇ hciγω}
(1.25)
However, the conditions under which this one-equation approximation is valid
remain unclear. This kind of simplification has not received a lot of attention. It has
been examined previously for the case of a stratified medium and an asymptotic
behavior Eq (1.26) was proved directly from the lower-scale mass balance equations
for stratified systems [166], and used extensively in [150] which introduced the word
Taylor’s dispersion. More generally, it has been suggested in [295] that some time
constraints can be formulated allowing this one-equation description of the transport
processes. They considered the time-infinite behavior of the first two centered spatial
moments to determine an expression for the effective velocity and the dispersion
tensor. The work by [65] in a reactive case suggests that this behavior is quite general,
that is, it seems that a multidomain decomposition of the medium is necessary
only to describe short time phenomena (at least when the spatial operators are
linear). Such simplifications in the modelization, as well as a minimalist description
of the dimensionality of the problem Eq (1.26), are extremely appealing from an
experimental point of view and are often used without extensive discussions.
(εγ + εω )
∂hciγω
∂hciγω
∂2 hciγω
+ V∞
= D∞ ∂x2
∂t
∂x
(1.26)
In this equation, we use V ∞ for the velocity in the one-dimensional situation (as
compared to V for the tensor) to avoid any confusion with future notations.
73
74
INTRODUCTION
O N E - E Q U AT I O N S P E C I A L D E C O M P O S I T I O N N O N - E Q U I L I B R I U M M O D E L
Apart from this two-equation model and its two particular behaviors, a one-equation
non-equilibrium theory Eq (1.27), based on a very distinct background, is also available in [205] for mass transport and in [178] for heat transfer. The method used for
the development of this model differs from the volume averaging with closure theory
and finds its essence in a peculiar perturbation decomposition. This one-equation
non-equilibrium model can also be obtained using the homogenization theory and
this has been devised in [178]. Numerical simulations in [205] suggest that both oneequation local non-equilibrium models Eq (1.25) and Eq (1.27) might be one and the
same. However, it is not straightforward from the expression of the dispersion tensors
D∞ and D∗ and their connection needs to be clarified.
∂ hciγω
(εγ + εω )
+ V · ∇ hciγω = ∇ · {D∗ · ∇ hciγω }
∂t
(1.27)
DISCUSSION
The counterpart of all these one-equation models is that they capture less physics
of the transport processes than two-equation models and, hence, there are some
stronger assumptions associated with Eqs (1.25), (1.26) and (1.27). For example, consider the mass transport within two tubes flowing at two very different mean velocities.
Following the work in [248] and later in [9], the description of this system can be undertaken at long times using two independent advection-dispersion equations and
not by a single one. Hence, it is important to develop an intelligible domain of validity
that can be used in applications. Unfortunately, the works aforementioned do not provide explicit representations of these assumptions, except for some elements in [249],
that can be used by experimenters. In most developments of this type, it is unclear
when it is possible to neglect the influence of higher order moments and what are
the temporal constraints associated with the time-infinite limits considered. In the
volume averaging developments, constraints focus on the characteristic times of the
fluctuations, and it is not always clear how this should be used to specify constraints
for experimental studies.
1.6
S C O P E O F T H I S PA RT
To summarize, we have the following models, classified decreasingly in relation to
the quantitity of information they capture from the microscale problem, and their
complexity thereof.
• A fully non-local formulation (Model A) Eq (1.5) which is an exact solution (in
terms of ensemble average not necessarily equivalent to volume average) but
often impractical.
1.6 S C O P E O F T H I S P A R T
Figure 27: Schematic representation of the transient behavior of a pulse evolving in a twophase/region porous medium.
• A two-equation transient closure model (Model B) Eqs (1.21-1.22) involving time
convolutions that account for time non-locally but approximates the spatial
convolutions, using effective parameters calculated on a REV.
• A two-equation quasi-stationary model (Model C) Eqs (1.19-1.20) that approximates the spatial convolutions similarly to Model B but also approximates the
time convolutions.
• A one-equation non-equilibrium model (Model D) Eqs (1.25) and (1.26) that
approximates both convolutions, under stronger temporal hypothesis than
Model C.
• A one-equation non-equilibrium model (Model E) Eq (1.27), obtained using a
special perturbation decomposition.
• A one-equation local mass equilibrium model (Model F) Eq (1.23) that imposes
strong physical limitations regarding the pointwise concentration within the
different phases.
The main contributions of this study are to
1. develop a comprehensive framework, based on the volume averaging with
closure theory, for the development of the Models B to F.
2. study the limitations associated to Model C in relation to Model B. This connection has already been discussed in [177, 237] using some rather intuitive
arguments. Herein, we study the analytical response of both models to sinusoidal excitations on a very simple stratified geometry. We show that the Model
C is valid if and only if the characteristic time for the relaxation of the effective
parameters is significantly smaller than the period of the input signal.
3. express the constraints that allow one to use Model D instead of Model C. For
the expression of these constraints, we will only consider the case of an infinite
75
76
INTRODUCTION
Figure 28: Schematic representation of the two methods that can be used for the development
of a one-equation non-equilibrium model.
one-dimensional porous medium in which the boundary/initial condition is
a Dirac distribution. There has been some discussion concerning the influence of the boundary conditions on the time-asymptotic regime [73] but we
will not consider such situations. This is particularly interesting to express
the inequalities that are used to describe the time restrictions only in terms
of the effective parameters of the two-equation quasi-stationary model. The
reason for this is that these are quite intuitive, that is, with some experience
one may apprehend these limitations really well. The situation is reminiscent
to the problem of defining a measure of Gaussian-like distribution in probabilistic analysis. Hence, our work is based on the expression of higher order
moments and especially of the standardized moments of order 3 and 4, that is,
respectively skewness and kurtosis. Physical interpretations of the constraints
are thoroughly discussed. The main idea is that, at long times, the spreading
dominates all other types of deformation of the signal, as illustrated on Fig (27).
4. study the convergence of the Model C toward the Model D in terms of spatial
moments.
5. advance a formal proof for the equivalence (Fig. 28) between Models D and E.
As an aside, it is interesting to emphasize that very few effort has been dedicated to establishing connections between the various upscaling techniques.
Noticeable exceptions are a comparison of ensemble averaging and volume
averaging [283], a comparison of the moment matching technique and the
homogenization theory [12], a comparison of the homogenization theory and
the volume averaging theory [31] and an equivalence between the continuous
time random walk technique and the volume averaging theory [186].
6. group these information on a domain of validity illustration. Establishing the
domains of validity of macroscopic models represents a critical issue for both
theoreticians and experimenters. The procedure of upscaling aims to reduce the
1.6 S C O P E O F T H I S P A R T
number of degrees of freedom of a problem by eliminating redundancy in the
information at the microscopic scale (see in [282]). It must be understood that
the quantity of information that must be eliminated depends on the problem
itself but also on the tolerance that one has toward the solutions. Hence, constraints are usually expressed in terms of orders of magnitude of dimensionless
numbers such as the Péclet number for example. For the dual-region situation,
the reader is referred to the discussion in [113] and more specifically to the
Fig. 2 which provides a very good description of the domains of validity of the
different models in terms of two Péclet numbers and the ratio between the two.
Herein, we show that a temporal axis must also be considered and we discuss
the effect of the boundary conditions on these domains. It is shown that for the
modelization of a pulse through a porous medium, one may first consider a
fully non-local model at small times, then the transient two-equation model,
the two-equation quasi-stationary formulation and at last the one-equation
form.
77
2
M AT H E M AT I C A L D E V E L O P M E N T S F O R T H E M A C R O T R A N S P O R T
T H E O RY
2.1
T W O - E Q U AT I O N
( MODELS
B AND C )
2.1.1 Volume averaging
To start, the averaging operators and decompositions defined above are applied to
Eqs (1.1)-(1.3); the details of this process are provided in Appendix A. The result is
γ − phase
∂εγ hcγ iγ
∂t
Z
Z
1
1
= ∇ · Dγ · ∇ (εγ hcγ iγ ) +
nγω cγ dS +
nγσ cγ dS
V S γω
V S γσ
Z
Z
1
1
+
nγω · Dγ · ∇cγ dS +
nγσ · Dγ · ∇cγ dS
V S γω
V S γσ
− ∇ · (εγ hvγ cγ iγ )
(2.1)
ω − phase
∂εω hcω iω
∂t
Z
Z
1
1
ω
= ∇ · Dω · ∇ (εω hcω i ) +
nωγ cω dS +
nωσ cω dS
V S ωγ
V S ωσ
Z
Z
1
1
+
nωγ · Dω · ∇cω dS +
nωσ · Dω · ∇cω dS
V S ωγ
V S ωσ
− ∇ · (εω hvω cω iω )
(2.2)
To make further progress, we decompose the concentrations as an average plus a
fluctuation Eqs (1.12)-(1.13) and assume that the hypothesis of spatial locality (separation of length scales) is valid (see in Appendix A). We show that the macroscopic
equation takes the form
γ − phase
79
80
M AT H E M AT I C A L D E V E L O P M E N T S F O R T H E M A C R O T R A N S P O R T T H E O R Y
∂εγ hcγ iγ
+
∂t
Z
Z
1
1
nγω c̃γ dS +
nγσ c̃γ dS
= ∇ · εγ Dγ · ∇hcγ iγ +
Vγ S γω
Vγ S γσ
Z
Z
1
1
nγω · Dγ · ∇c̃γ dS +
nγσ · Dγ · ∇c̃γ dS − ∇ · (εγ hc̃γ ṽγ i)
+
V S γω
V S γσ
− ∇ · (εγ hcγ iγ hvγ iγ )
(2.3)
ω − phase
∂εω hcω iω
∂t
Z
Z
1
1
nωγ c̃ω dS +
nωσ c̃ω dS
= ∇ · εω Dω · ∇hcω iω +
Vω S ωγ
Vω S ωσ
Z
Z
1
1
nωγ · Dω · ∇c̃ω dS +
nωσ · Dω · ∇c̃ω dS − ∇ · (εω hc̃ω ṽω i)
+
V S ωγ
V S ωσ
− ∇ · (εω hcω iω hvω iω )
(2.4)
Equations (2.3)-(2.4) represent a macroscopic description of the transport processes. However, the problem is not under a closed form, that is, in addition to terms
involving hci ii there are terms involving c̃i . In order to close the problem, we need (1)
to determinate the boundary value problems that the perturbations satisfy and (2)
to uncouple, in these problems, the contributions varying at the macroscale (source
terms) from those varying at the microscale.
2.1.2 Fluctuations equations
Going back to their definition c̃i = ci − hci ii suggests that the set of equations governing the behavior of these perturbations can be obtained by subtracting Eqs (2.1)-(2.2)
to, respectively, Eqs (1.1)-(1.3). This operation, in conjunction with the previous averaging process, leads to two different set of equations. One governs the macroscopic
concentrations Eqs (2.1)-(2.2) or Eqs (2.3)-(2.4). The other, the results of the operations (1.1 minus 2.1) and (1.3 minus 2.2), describes the perturbations. A priori, these
are coupled and must be solved simultaneously, that is, we end up with a non-local
description of the transport. Under the assumption that a local solution is desired,
the coupling can be broken down by assuming a separation of length scales. Hence,
we use (1.1 minus 2.3) and (1.3 minus 2.4) instead of (1.1 minus 2.1) and (1.3 minus
2.2) to describe the behavior of the perturbations. For the same reason, we assume
that derivatives of macroscopic concentrations can be neglected in comparison to
derivatives of fluctuations. For a more detailed discussion, the reader is referred to
the work of Quintard and Whitaker in [207, 208, 209, 210, 211]. This operation leads
to
2.1 T W O - E Q U AT I O N ( MODELS B AND C )
81
∂c̃γ
+ ∇ · (c̃γ vγ ) − h∇ · (c̃γ vγ )iγ + ṽγ .∇hcγ iγ = ∇ · {Dγ · ∇c̃γ } − h∇ · {Dγ ∇c̃γ }iγ
∂t
(2.5)
−nγσ · (Dγ · ∇c̃γ ) = nγσ · (Dγ · ∇hcγ iγ )
BC1 :
γ
c̃γ − c̃ω = − (hcγ i − hcω i )
nγω · jc̃γ − jc̃ω = −nγω · jhcγ iγ − jhcω iω
BC2 :
BC3 :
BC4 :
ω
−nωσ · (Dω · ∇c̃ω ) = nωσ · (Dω · ∇hcω i )
ω
on S γσ
(2.6a)
on S γω
(2.6b)
on S γω
(2.6c)
on S ωσ
(2.6d)
∂c̃ω
+ ∇ · (c̃ω vω ) − h∇ · (c̃ω vω )iω + ṽω .∇hcω iω = ∇ · {Dω · ∇c̃ω } − h∇ · {Dω ∇c̃ω }iω
∂t
(2.7)
D
where jΦi = − i ·∇Φi . At this point, the problem on the perturbations is still coupled
with the macroscopic concentrations but in a weaker sense. Using the superposition
principle for linear operators, in conjunction with the separation of the length scales,
one can perform the following change in variables, without any additional assumption
[178, 237].
c̃γ = bγγ · ∗
∂
∂
∂
∇hcγ iγ + bγω · ∗ ∇hcω iω − rγ ∗ (hcγ iγ − hcω iω )
∂t
∂t
∂t
c̃ω = bωγ · ∗
∂
∂
∂
∇hcγ iγ + bωω · ∗ ∇hcω iω − rω ∗ (hcγ iγ − hcω iω )
∂t
∂t
∂t
(2.8)
(2.9)
The ∗ operator refers to convolutions in time defined by
Zt
e (t) ∗ f (t) = e (τ) f(t − τ)dτ
(2.10)
0
The b and r fields can be interpreted as integrals of the corresponding Green’s functions when the spatial locality is assumed (see discussions in [282]). The goal of this
expression is to find variables that can be calculated locally on a REV.
In addition, we will assume that the medium can be represented locally by a periodic boundary condition. This concept concerning the notion of representative
elementary volume (REV) is widely misunderstood. In hierarchical porous media,
under the assumption that a REV can be defined, the information that is necessary
to calculate the effective parameters of the models is contained in a relatively small
82
representative portion of the medium. For this representative volume, there are two
type of boundary conditions. The first one concerns the different regions/phases
inside the REV ans is determined by the physics of the processes at the microscale.
The second type refers to the external boundary condition between the REV and the
rest of the porous medium. This condition is not determined by the physics at the
small scale but rather represents a closure of the problem. At first, it is unclear how
this choice should be made and it results in a significant amount of confusion in the
literature. From a theoretical point of view, if the REV is large enough, it has been
shown [274] that effective parameters do not depend on this boundary condition.
In real situations, this constraint is never exactly satisfied, that is, this boundary
condition has an impact on the fields. However, in the macroscopic equations, the perturbations appear only under integrated quantities. Because of this, the dependence
of effective parameters upon the solution of the closure problem is essentially mathematically of a weak form [190]. Hence, one could choose, say, Dirichlet, Neuman,
mixed or periodic boundary conditions to obtain a local solution which produces
acceptable values for the associated averaged quantities. The periodic boundary
condition lends itself very well for this application as it induces very little perturbation in the local fields, in opposition to, say, Dirichlet boundary conditions (this is
particularly useful for non-isotropic unit cell characteristics). It must be understood
that this does not mean that the medium is interpreted as being physically periodic.
For the remainder of this work, we will assume that the medium can be represented
locally by a periodic cell and that the effective parameters can be calculated over this
representative part of the medium.
2.1.3 Closure problems
Upon substituting these expressions for the fluctuations in Eqs (2.5) to (2.7), we
can collect separately terms involving ∇hcγ iγ , ∇hcω iω and hcγ iγ − hcω iω . This is
based on the assumption that the fluctuations can be decomposed on the basis of
{hcγ iγ , hcω iω , ∇hcγ iγ , ∇hcω iω }, that is, higher order derivatives can be neglected.
Collecting terms involving hcγ iγ − hcω iω leads to Eqs (2.11) to (2.13)
∂rγ
+ vγ · ∇rγ = ∇ · {Dγ · ∇rγ } − ε−1
γ h (t)
∂t
−nγσ · (Dγ · ∇rγ ) = 0
BC1 :
BC2 :
BC3 :
BC4 :
D
rγ − rω = 1
Dω · ∇rω) = 0
−nωσ · (Dω · ∇rω ) = 0
nγω · ( γ · ∇rγ −
Periodicity : ri (x + pk ) = ri (x)
(2.11)
on S γσ
(2.12a)
on S γω
(2.12b)
on S γω
(2.12c)
on S ωσ
(2.12d)
k = 1, 2, 3
(2.12e)
∂rω
+ vγω · ∇rω = ∇ · {Dω · ∇rω } + ε−1
ω h (t)
∂t
(2.13)
where, using previous assumptions, we can write
h (t) =
=
1
V
Z
S γω
Dγ∇rγ}iγ =Z −εωh∇ · {Dω∇rω}iω
1
nγω · Dγ · ∇rγ dS = −
nωγ · Dω · ∇rω dS
V
εγ h∇ · {
(2.14)
S γω
We have used pk to represent the three lattice vectors that are needed to describe
the 3-D spatial periodicity.
Collecting terms involving ∇hcγ iγ leads to Eqs (2.15) to (2.17)
∂bγγ
∗
+ vγ · (∇bγγ − rγ I) + ṽγ = ∇ · {Dγ · (∇bγγ − rγ I)} − hṽγ rγ iγ − ε−1
γ β1
∂t
(2.15)
BC1 :
−nγσ · (Dγ · ∇bγγ ) =nγσ · Dγ
BC2 :
BC3 :
BC4 :
on S γσ
(2.16a)
bγγ − bωγ =0
−nγω · Dγ · (∇bγγ − rγ I) =nγω · Dγ − nγω · Dω · (∇bωγ − rω I)
−nωσ · (Dω · ∇bωγ ) =0
on S γω
(2.16b)
on S γω
(2.16c)
on S ωσ
(2.16d)
Periodicity : bij (x + pk ) =bij (x)
k = 1, 2, 3
(2.16e)
∂bωγ
∗
+ vγ · (∇bωγ − rω I) = ∇ · {Dω · (∇bωγ − rω I)} − hṽω rω iω + ε−1
ω β1
∂t
(2.17)
where
D
D
β∗1 = h∇ · { γ · (∇bγγ − rγ I)}i = −h∇ · { ω · (∇bωγ − rω I)}i
(2.18)
Z
Z
1
1
=
nγω · γ · (∇bγγ − rγ I) dS = −
nωγ · ω · (∇bωγ − rω I) dS
V Sγω
V Sγω
D
D
(2.19)
Collecting terms involving ∇hcω iω leads to Eqs (2.20) to (2.22)
83
84
∂bγω
∗
+ vγ · (∇bγω + rγ I) = ∇ · {Dγ · (∇bγω + rγ I)} + hṽγ rγ iγ − ε−1
γ β2
∂t
BC1 :
−nγσ · (Dγ · ∇bγω ) = 0
BC2 :
bγω − bωω = 0
BC3 :
BC4 :
nγω ·
on S γσ
(2.21a)
Dγ · (∇bγω + rγ I) = nγω · Dω + nγω · Dω · (∇bωω + rω I)
−nωσ · (Dω · ∇bωγ ) = nωσ · Dω
on S γω
(2.21b)
on S γω
(2.21c)
on S ωσ
(2.21d)
bij (x + pk ) =bij (x)
Periodicity :
(2.20)
k = 1, 2, 3
(2.21e)
∂bωω
∗
+ vω · (∇bωω + rω I) + ṽω = ∇ · {Dω · (∇bωω + rω I)} + hṽω rω iω + ε−1
ω β2
∂t
(2.22)
where
D
D
β∗2 = h∇ · { γ · (∇bγω + rγ I)}i = −h∇ · { ω · (∇bωω + rω I)}i
(2.23)
Z
Z
1
1
=
nγω · γ · (∇bγω + rγ I) dS = −
nωγ · ω · (∇bωω + rω I) dS
V Sγω
V S γω
D
D
(2.24)
The unicity of the fields is provided by
hrγ iγ = 0, hrω iω = 0, hbγγ iγ = 0, hbγγ iγ = 0, hbγω iγ = 0, hbωγ iω = 0, hbωω iω = 0
(2.25)
rγ (t = 0) = rω (t = 0) = 0, bγγ (t = 0) = bγω (t = 0) = bωγ (t = 0) = bωω (t = 0) = 0
(2.26)
Both problems are coupled together but uncoupled from the macroscopic problems Eqs (2.3)-(2.4). Hence, these fields can be calculated locally on a REV. However,
fluctuations still remain in Eqs (2.3)-(2.4) and it is now necessary to uncouple the
macroscopic equations from the perturbations.
2.1.4 Macroscopic equations Model B
In order to get rid of these perturbations, we inject Eqs (2.8)-(2.9) in Eqs (2.3)-(2.4).
The result can be expressed by Eqs (2.27)-(2.28) [178, 237]
εγ
εω
∂hcγ iγ
∂t
∂hcω iω
∂t
∂
∂
+ εγ Vγγ . ∗ ∂t
∇hcγ iγ + εγ Vγω . ∗ ∂t
∇hcω iω = εγ ∇. {Dγγ . ∗
∂
∇hcγ iγ }
∂t
∂
+εγ ∇. {Dγω . ∗ ∇hcω iω }
∂t
∂
γ
−h ∗ (hcγ i − hcω iω )
∂t
+εγ Qγ (x, y, z, t)
(2.27)
∂
∂
+ εω Vωγ . ∗ ∂t
∇hcγ iγ + εω Vωω . ∗ ∂t
∇hcω iω = εω ∇. {Dωγ . ∗
∂
∇hcγ iγ }
∂t
∂
+εω ∇. {Dωω . ∗ ∇hcω iω }
∂t
∂
−h ∗ (hcω iω − hcγ iγ )
∂t
+εω Qω (x, y, z, t)
(2.28)
Herein, the macroscopic boundary conditions are described by the source terms
distributions εγ Qγ (x, y, z, t) and εω Qω (x, y, z, t). Effective velocities, containing intrinsic averages of the microscale velocities, are
∗
γ
Vγγ = hvγ iγ − ε−1
γ β1 − hṽγ rγ i
∗
ω
Vωω = hvω iω + ε−1
ω β2 + hṽω rω i
(2.29)
(2.30)
Interphase coupling arising from the interfacial flux as well as from fluctuations of
the velocities can be written
∗
γ
Vγω = −ε−1
ω β2 + hṽγ rγ i
∗
ω
Vωγ = ε−1
γ β1 − hṽω rω i
(2.31)
(2.32)
For dispersion effects, dominant terms are
Dγγ
=
D
Dωω
=
D
Z
1
nγω bγγ dS − hṽγ bγγ iγ
γ I+
Vγ Sγω
Z
1
nωγ bωω dS − hṽω bωω iω
ω I+
Vω Sγω
(2.33)
(2.34)
85
86
Interphase dispersion couplings are
Dγω
=
D
Dωγ
=
D
Z
1
nγω bγω dS − hṽγ bγω iγ
γ
Vγ Sγω
Z
1
nωγ bωγ dS − hṽω bωγ iω
ω
Vω Sγω
(2.35)
(2.36)
and the first-order exchange coefficient
1
h=
V
Z
D
1
nγω · γ · ∇rγ dS = −
V
S γω
Z
S γω
nωγ ·
Dω · ∇rωdS
(2.37)
With these definitions, it must be understood that all the effective parameters
exhibit a time-dependence, even though we have used ij , Vij and h instead of
ij (t), Vij (t) and h (t), to simplify the notations.
D
D
2.1.5 Local Model C and conditions for time-locality
εγ
∂hcγ iγ
+ εγ Vγγ (∞) .∇hcγ iγ + εγ Vγω (∞) .∇hcω iω
∂t
= εγ ∇. {Dγγ (∞) .∇hcγ iγ }
+εγ ∇. {Dγω (∞) .∇hcω iω }
−h (∞) (hcγ iγ − hcω iω )
+εγ Qγ (x, y, z, t)
εω
∂hcω iω
+ εω Vωγ (∞) .∇hcγ iγ + εω Vωω (∞) .∇hcω iω
∂t
(2.38)
= εω ∇. {Dωγ (∞) .∇hcγ iγ }
+εω ∇. {Dωω (∞) .∇hcω iω }
−h (∞) (hcω iω − hcγ iγ )
+εω Qω (x, y, z, t)
(2.39)
Effective parameters, such as h (t), undertake a transient regime and then reach a
stationary state, that is, after a given relaxation time τ1 , h (t) tends toward a constant
h (∞) . To understand when the local two-equation model Eqs (2.38)-(2.39) represents
a good approximation of Eqs (2.27)-(2.28) boils down to understanding when
∇·{
Dij (t) · ∗ ∂t∂ ∇hcjij (t)} can be approximated by ∇ · {Dij (∞) · ∇hcjij (t)}
Vij (t) · ∗
∂
∇hcj ij (t) by Vij (∞) · ∇hcj ij (t)
∂t
h (t) ∗
∂
(hcγ iγ (t) − hcω iω (t)) by h (∞) (hcγ iγ (t) − hcω iω (t))
∂t
It is understood that the convolutions converge toward the local formulation Eqs
(2.38)-(2.39) when the effective parameters relax quickly in comparison to characteristic times for macroscopic variations. Under this condition, for any effective
parameter κ and any function f, the convolution can be approximated the following
way [237, 177]
∂
∂
κ (t) ∗ f (t) ≈ κ (∞) ∗ f (t) =
∂t
∂t
Zt
κ (∞)
0
∂
f (τ) dτ = κ (∞) f (t)
∂τ
(2.40)
To illustrate this asymptotic behavior, we consider a simplified configuration for
which an analytical solution is available. Explicitly, we are interested in the following
microscale diffusion problem
∂2 cmicro
∂cmicro
γ
γ
= Dγ
∂t
∂x2
(2.41)
with −l 6 x 6 +l, zero initial concentration and the boundary conditions x = ±l
maintained at concentration sin (ωt) for t > 0 (such a physical situation is purely
theoretical for mass transport but could be realizable for heat transfer problems). The
idea behind the use of a sinusoidal input is that the macroscopic signal is charac1
terized by a single time, say T = ω
. We define the macroscopic averaged quantities
as
1
hφγ i =
2l
Z +l
γ
φγ dx
(2.42)
−l
The averaged mass balanced equations can be written
dhc∗γ iγ
d
dhc∗ iγ
+h (t) ∗ dtγ = h ∗ sin (ωt)
εγ
dt
dt
(2.43)
and
εγ
dhcγ iγ
+h (∞) hcγ iγ = h sin (ωt)
dt
(2.44)
in which we have added the ∗ superscript to differentiate between the non-local
and the local formulations. It is fundamental to notice that, in this very particular
configuration, the constraints associated with Eq (2.43) are exactly satisfied, that is,
87
88
hc∗γ iγ = hcmicro
iγ . In our case, for obvious symmetry reasons, hcmicro
iγ does not vary
γ
γ
along the z axis. Hence, the spatial locality is exactly verified and the closure on the
perturbations is an exact solution.
Therefore, we only need to compare hcγ iγ with hc∗γ iγ , in order to understand the
behavior of the local model in relation to the non-local formulation. Toward this
goal, we first determine the microscale solution of Eq (2.41), and calculate hc∗γ iγ =
hcmicro
iγ . As a second step, we solve Eq (2.44) to determine hcγ iγ . At last, we develop a
γ
characteristic time τ1 for the relaxation of h and compare the long-time macroscopic
τ
results for various ratio T1 .
Expression of the pointwise microscopic solution and hc∗γ iγ = hcmicro
iγ
γ
The analytical microscopic solution of the problem can be written [39] page (105)
cmicro
γ
= Asin (ωt + Φ)
+16l2 ωπDγ
∞
X
(2.45)
(−1)n (2n + 1)
4 2
2 4
n=0 16l ω + Dγ π (2n + 1)
4
cos
(2n + 1) πx −Dγ (2n+1)2 π2 t/4l2
e
2l
with
Φ = arg {
r
A=
r
k=
cosh kx (1 + i)
}
cosh kl (1 + i)
(2.46)
cosh 2kx + cos 2kx
cosh 2kl + cos 2kl
(2.47)
ω
2Dγ
(2.48)
Hence, at long times, we have
hc∗γ iγ =hcmicro
iγ = hAsin (ωt + Φ)iγ
γ
t→∞
(2.49)
Local solution hcγ iγ
Using Laplace transforms on Eq (2.44), we obtain
h
εγ
h
ε ω εγ
−ht
γ
hcγ i = r sin ωt − arctan
+ 2
e εγ (2.50)
2
h
h
h
+ ω2
+ ω2
εγ
γ
εγ
and at long times
hcγ iγ =
t→∞
r
h
εγ
sin
2
h
2
+ω
εγ
ωt − arctan
εγ ω h
(2.51)
Relaxation of h
The determination of a characteristic time τ1 for the relaxation of h is actually the
most difficult part. Going back to its definition, Eq (2.37), we have
1
h=
2l
Z +l
Dγ
−l
∂rγ
dx
∂x
(2.52)
with rγ solution of the following boundary value problem
∂rγ
∂2 rγ
= Dγ 2 − ε−1
γ h
∂t
∂x
BC :
(2.53)
rγ = 1
on S γω
(2.54)
We also have
hrγ iγ = 0
(2.55)
One way around the calculation of h is to decompose rγ using
rγ = 1 − Rγ ∗
∂
h
∂t
Hence, we have Rγ solution of (straightforward in Laplace space)
∂Rγ
∂2 Rγ
= Dγ
+ ε−1
γ
∂t
∂x2
BC :
(2.56)
Rγ = 0
on S γω
(2.57)
89
90
The solution of Equation (2.56) is [39] p. 130
∞
(2n + 1) πx −Dγ (2n+1)2 π2 t/4l2
l2
x2 32 X (−1)n
{1 − 2 − 3
} (2.58)
cos
Rγ =
e
3
2εγ Dγ
2l
l
π
(2n + 1)
n=0
and the expression of h is derived from Equation (2.55)
hRγ iγ (t) ∗
∂
h (t) = 1
∂t
(2.59)
A characteristic time for the asymptotic relaxation of h corresponds to the long2
γ
time relaxation of hRγ i , that is, τ1 = π4l
2 D . By definition, the h (∞) appearing in
γ
Equation (2.44) corresponds to the stationary part of Equation (2.59), that is,
h (∞) =
3εγ Dγ
l2
(2.60)
τ
Comparison of hcmicro
iγ and hcγ iγ in the long-time limit for various ratio T1
γ
We fix l = 0.5, εγ = 0.6 and the time t is normalized in relation to a characteristic time
for the oscillations, that is, t 0 = Tt = ωt. We compare the results for different ratio
τ1
T
2
= π4l
2 D ω in the long time regime, that is, all the exponentially decaying terms are
γ
not considered. We integrate and plot Equation (2.61) using MAPLET M (Micro on
Figure 29 )
iγ ≈ hAsin (ωt + Φ)iγ
γ
(2.61)
and we compare it to (Local on Figure 29 )
h
εγ
γ
hcγ i = r
h
εγ
2
+ ω2
ε ω γ
sin ωt − arctan
h
(2.62)
It is clear from Fig (29) that we require τ1 T in order for the Eq (2.43) to converge
toward Eq (2.44), that is, the characteristic time of the macroscopic signal needs to be
much larger than the characteristic time for the relaxation of the exchange coefficient
h. In a more general context, this suggests that the Eqs (2.38)-(2.39) represent a good
approximation of Eqs (2.27)-(2.28) under the condition that T τ1 , in which τ1 can
be estimated using
τ1 = O
l2γ l2ω
lγ
lω
}
max {
,
,
,
hvγ iγ hvω iω Dγ Dω
!
(2.63)
τ
τ
a) T1 = 0.01
b) T1 = 0.1
τ
τ
c) T1 = 1
d) T1 = 100
γ
γ
i = hc∗γ i (Non-local/Micro) and
Figure 29: Comparison of the responses hcmicro
γ
hcγ iγ (Local), plotted as functions of Tt , for various ratio τT1 (εγ = 0.6, l = 0.5).
91
92
and
T = O min {Thcγ iγ , Thcω iω , T∇hcγ iγ , T∇hcω iω , T∇2 hcγ iγ , T∇2 hcω iω }
(2.64)
where Tφ is the characteristic time associated with time variations of φ. These are
related to (1) the macroscopic processes but also to (2) the boundary condition, as
illustrated in the previous sinusoidal example.
Can we make a “better” two-equation model ?
There is an interesting idea that would consist in adapting the exchange coefficient
h to the oscillations, that is, developing h as a function of ω. The purpose of this
section is to investigate the pertinence of such a method.
As a reminder, we have
hc∗γ iγ =hcmicro
iγ = hAsin (ωt + Φ)iγ
γ
t→∞
(2.65)
with
Φ = arg {
r
A=
r
k=
cosh kx (1 + i)
}
cosh kl (1 + i)
(2.66)
cosh 2kx + cos 2kx
cosh 2kl + cos 2kl
(2.67)
ω
2Dγ
(2.68)
and
hcγ
iγ
=
t→∞
r
h(ω)
εγ
h(ω) 2
εγ
+ω2
εγ ω
sin ωt − arctan h(ω)
(2.69)
There is a decomposition of hc∗γ iγ Eq. (2.70) that considerably helps in the comparison of both signals.
iγ = hAsin (ωt + Φ)iγ = Csin(ωt + D)
γ
t→∞
(2.70)
2.2 O N E - E Q U AT I O N L O N G - T I M E B E H AV I O R ( MODEL D )
where
q
C=
2
(hAsin (Φ)iγ ) + (hAcos (Φ)iγ )
2
(2.71)
and
hAsin (Φ)iγ
D = arctan
hAcos (Φ)iγ
(2.72)
Hence, we would need both amplitudes and phases to be equal, that is,
hamplitude (ω)
εγ
r
hamplitude (ω)
εγ
2
q
=
2
(hAsin (Φ)iγ ) + (hAcos (Φ)iγ )
2
(2.73)
+ ω2
and
εγ ω
−arctan
hphase (ω)
hAsin (Φ)iγ
≡ arctan
hAcos (Φ)iγ
(2.74)
This can be expressed by
q
2
2
εγ ω (hAsin(Φ)iγ ) +(hAcos(Φ)iγ )
hamplitude (ω)= q
2
2
1−(hAsin(Φ)iγ ) −(hAcos(Φ)iγ )
(2.75)
and
ε ωhAcos(Φ)iγ
hphase (ω) ≡- γhAsin(Φ)iγ
(2.76)
Therefore, it seems that h is overdetermined by this system. Using a formulation
with an exchange coefficient h (ω) that depends on the frequency of the input, it looks
like it is not possible to recover the phase and the amplitude simultaneously. On Fig
(30), we plotted both hamplitude and hphase and showed that these two expressions
only overlap in the low frequency limit.
2.2
O N E - E Q U AT I O N L O N G - T I M E B E H AV I O R
( MODEL
D)
We are now interested in a long time simplification of this two-equation local model.
It has been shown in [295] that the system Eqs (2.38)-(2.39) has a time-asymptotic
93
94
2
Figure 30: Representations of hphase and hamplitude as functions of τT1 = ω π4l
2D .
γ
behavior for semi-infinite or infinite systems which can be described in terms of a
one equation advection-diffusion type model Eq (2.77).
∂ hciγω
(εγ + εω )
+ V · ∇ hciγω = ∇ · {
∂t
D∞ · ∇ hciγω}
(2.77)
The analysis is based on the study of the long time behavior of the first two centered
spatial moments. There is an interesting discussion concerning higher order moments
which has not been tackled yet. One usually assumes that the behavior of centered
spatial moments of order superior to three can be neglected. From the analysis
conducted in [295], it is emphasized that, in a dual-phase situation, the distance
between the signals in each phase tends toward a constant Ω at long times Fig (27).
Thus, the signal hciγω can not be exactly Gaussian shaped at long times, that is,
there is no convergence in terms of centered moments. However, we show that we
have a convergence in terms of these moments standardized to the second order
centered moment. In essence, it means that in the time-asymptotic regime, the
spreading dominates all other forms of deformation of the spatial signal. This idea
is reminiscent to the Taylor-Aris dispersion in a tube. Herein, we provide a rigorous
analysis in which (1) we develop constraints for neglecting of higher order moments
and (2) we study the convergence toward the asymptotic regime. We show that the
concentration fields asymptotically tend toward a Gaussian shape signal in space
described by Eq (2.78)
∂hciγω
∂hciγω
∂2 hciγω
+ V∞
= D∞ ∂x2 + Qγω
∂t
∂x
(2.78)
with Eq (2.79)
hciγω =
εγ
εω
hcγ iγ +
hcω iω
εγ + εω
εγ + εω
(2.79)
and Eq (2.80)
Qγω =
εγ
εω
Qγ +
Qω
εγ + εω
εγ + εω
(2.80)
The following are our main hypotheses, and are thoroughly discussed along the
demonstration:
HYPOTHESIS 1
Our work is limited to a 1-D infinite medium in which we assume
that there is no mass in the system at t = 0 and the input is a Dirac signal.
HYPOTHESIS
2
We will only consider the case Qγ (t) = Qω (t) = Qγω δ(t).
HYPOTHESIS 3
We make the following approximations Vγω − Vωγ Vγγ − Vωω
and Dγω − Dωγ Dγγ − Dωω .
Our strategy consists in using various mathematical objects, known as spatial moments, that measure some fundamental aspects of the spatial distribution of the
concentration. We start with the study of the simple raw spatial moments and show
that, using Laplace space formulations and mathematical analyses of the boundary
value problem, the dominant term at long times captures an information regarding
the asymptotic velocity of propagation of the signal. To recover information regarding
the spreading of the signals, we define centered moments and formulate an expression of the nth order centered moments based on an extrapolation from the first
centered moments. The goal is to compare these to the centered moments of the
one-equation model Eq. (2.78) that are explicitly presented in section 2.2.3. We prove
that there is not an asymptotic convergence of the two-equation model toward a oneequation model in terms of central moments. However, we show that there is a weaker
convergence in terms of standardized moments and this is discussed in section 2.2.4.
Eventually, we propose a measure defined in terms of a relative difference of the
standardized moments between the two-equation and the one-equation models. We
also thoroughly discuss sufficient conditions for the convergence in section 2.2.5.
95
96
2.2.1 Raw moments analysis
To start our mathematical analysis, we define the raw spatial moments in the i −
phase by Eq (2.81)
+∞
Z
µin (t) =
−∞
xn hci ii dx
(2.81)
First, our goal is to study the temporal behavior of these raw moments. Toward
that end, we derive the moments generating differential equations Eqs (2.82)-(2.83).
There are, at least, two ways to develop these equations. The first one is based on the
Fourier transform moments generating function and the second consists in a direct
integration by part of the equations (2.38)-(2.39), for n > 2
∂µγ
n
∂t
= n (n − 1)
Dγγ µγn−2 + n (n − 1) Dγω µωn−2 + nVγγ µγn−1 + nVγω µωn−1
ω
− εhγ (µγ
n − µn )
∂µω
n
∂t
= n (n − 1)
(2.82)
Dωγ µγn−2 + n (n − 1) Dωω µωn−2 + nVωγ µγn−1 + nVωω µωn−1
γ
− εhω (µω
n − µn )
(2.83)
with
∂µγ0
h γ
=−
µ0 − µω
0
∂t
εγ
(2.84)
∂µω
h
γ
0
=−
µω
0 − µ0
∂t
εω
(2.85)
∂µγ1
h γ
ω
= Vγγ µγ0 + Vγω µω
µ
−
µ
−
0
1
∂t
εγ 1
(2.86)
∂µω
h
γ
1
= Vωγ µγ0 + Vωω µω
µω
0 −
1 − µ1
∂t
εω
(2.87)
and
It is straightforward to show that
(Qγ (t) − Qω (t)) ∗ e−αt
µ0 =
(εγ + εω )
"
εω
#
"
+ Qγω
−εγ
1
#
(2.88)
1
where α = h ε1γ + ε1ω . For the sake of simplicity, we will only consider the case
" #
Qγ (t) = Qω (t) = Qγω δ(t), which leads to µ0 = Qγω
1
1
. This assumption is not
fundamental but leads to substantial simplifications.
To make further progress, we consider the Laplace
transforms of the moments defined by Eq (2.89)
L A P L A C E S P A C E F O R M U L AT I O N
+∞
Z
µ̄in
e−pt µin (t) dt
=
(2.89)
−∞
The Laplace transform of the equations (2.82)-(2.83) leads to Eqs (2.90) and (2.91)
pµ̄γ
n
= n (n − 1)
Dγγ µ̄γn−2 + n (n − 1) Dγω µ̄ωn−2 + nVγγ µ̄γn−1 + nVγω µ̄ωn−1
ω
− εhγ (µ̄γ
n − µ̄n )
pµ̄ω
n
= n (n − 1)
(2.90)
Dωγ µ̄γn−2 + n (n − 1) Dωω µ̄ωn−2 + nVωγ µ̄γn−1 + nVωω µ̄ωn−1
γ
− εhω (µ̄ω
n − µ̄n )
(2.91)
which can be set under a matricial form Eq (2.92)
"
p + εhγ
− εhγ
− εhω
p + εhω
#"
µ̄γn
µ̄ω
n
#
"
= n (n − 1)
"
+n
Vγγ
Dγγ Dγω
Dωγ Dωω
Vγω
Vωγ Vωω
#"
#"
µ̄γn−1
µ̄ω
n−1
µ̄γn−2
#
µ̄ω
n−2
#
(2.92)
We have for n > 2, Eq (2.93)
µ̄n = n (n − 1) A−1 Dµ̄n−2 + nA−1 Vµ̄n−1
(2.93)
97
98
and Eqs (2.94)-(2.95) to complete the sequence
1
µ̄0 =
p
"
Qγω
#
(2.94)
Qγω
µ̄1 = A−1 Vµ̄0
(2.95)
with Eq (2.96)
"
A=
p + εhγ
− εhγ
− εhω
p + εhω
#
(2.96)
with Eq (2.97)
A−1 =
p2 + ph
1
1
εγ
"
+ ε1ω
p + εhω
h
εγ
h
εω
p + εhγ
#
(2.97)
1
α
I+
Λε
p+α
p (p + α)
1
1
(I − Λε ) + Λε
=
p+α
p
"
#
"
#
εγ εω
µ̄γn
ε
1
1
and with α = h εγ + εω , Λ =
, µ̄n =
,
εγ εω
µ̄ω
n
"
#
Vγγ Vγω
and V =
.
Vωγ Vωω
=
D=
"
Dγγ Dγω
Dωγ Dωω
#
G E N E R A L E X P R E S S I O N I N L A P L A C E S P A C E A N D L O N G T I M E L I M I T O F T H E R AW
S P AT I A L M O M E N T S
It is straightforward to show that, for n > 1,
"
µ̄n
µ̄n−1
#
"
=n
A−1 V (n − 1) A−1
1
n I2
02
D
#"
µ̄n−1
µ̄n−2
#
(2.98)
and that we have Eq (2.99)
"
#
µ̄n
p
= n!
µ̄n−1
k=0
Y
k=n−1
"
A−1 V kA−1
1
k+1 I2
D
02
#! "
A−1 02
02
02
#"
Q
0
#
(2.99)
We adopt a block matrix notation with Eqs (2.100), (2.101), (2.102), (2.103) and
(2.104)
A−1 V =
1
1
(V − Λε V) + Λε V
p+α
p
D
1
1
( − Λε ) + Λε
p+α
p
A−1
"
02 =
"
Q=
0 0
D
D
1 0
(2.102)
#
(2.103)
0 1
Qγω
(2.101)
#
0 0
"
I2 =
D
=
(2.100)
#
(2.104)
Qγω
Although Eq (2.99) represents a convenient formulation of the moments generating
differential equations, it is still quite complex. The dominant term at long times can
be extracted using the following final value theorem Eq (2.105)
lim f(t) = lim pf̄(p)
t→∞
(2.105)
p→0
Multiplying Eq (2.99) by pn and taking the limit when p → 0 leads to Eq (2.106), for
n > 1.
γ
ω
∞ n γω
lim (pµ̄γn ) = lim (pµ̄ω
+ O tn−1
n ) = lim (µn ) = lim (µn ) = n! (V t) Q
p→0
p→0
t→∞
t→∞
(2.106)
99
100
where we have Eq (2.107)
V∞ =
εγ (Vγγ + Vγω ) + εω (Vωγ + Vωω ) εγ hvγ iγ + εω hvω iω
=
εγ + εω
εγ + εω
(2.107)
The dominant term corresponds in fact to the moments of the following transport
equations
∂hcγ iγ
+ V ∞ ∇hcγ iγ = Qγω
∂t
(2.108)
∂hcω iω
+ V ∞ ∇hcω iω = Qγω
∂t
(2.109)
This term contains the information about the asymptotic velocity of propagation
of the signal within the 1-D infinite medium, but looses information concerning
the spreading. In this context, it is more interesting to consider the centered spatial
moments, that is we follow the signal on a frame moving at the velocity V ∞ . The
centered moments are defined by Eq (2.110).
+∞
Z
min =
x − µi1
−∞
n
hci ii dx
(2.110)
2.2.2 First centered moments and convergence
Herein, we used MapleT M to solve analytically the differential equations Eqs (2.82)
and (2.83). We adopt a general notation exp for all the exponentially decaying terms.
Results for the first order moments normalized to the total mass can be found in Eqs
(2.111) and (2.112)
mγ1
εω 1
(∆V + Vγω − Vωγ ) + V ∞ t + exp
=
γω
Q
εγ + εω α
|
{z
}
(2.111)
mω
εγ 1
1
(∆V + Vγω − Vωγ ) + V ∞ t + exp
=−
γω
Q
εγ + εω α
|
{z
}
(2.112)
εω
Cγ
1 = εγ +εω Ω
ε
γ
Cω
1 = εγ +εω Ω
with
Ω=
1
(∆V + Vγω − Vωγ )
α
(2.113)
and
∆V = Vγγ − Vωω
(2.114)
.
The constant Ω appearing in the first order moments represents the shift Fig (27)
between the two signals.
For the second order moments, we have Eqs (2.115) and (2.116).
γ
1 m2
= Cγ2 + D∞ t + exp
2 Qγω
(2.115)
1 mω
∞
2
= Cω
2 + D t + exp
γω
2Q
(2.116)
with Eq (2.117)
D∞
=
εγ (Dγγ + Dγω ) + εω (Dωγ + Dωω )
εγ + εω
1
(∆V + Vγω − Vωγ ) εγ εω ∆V − ε2γ Vγω + ε2ω Vωγ
+
2
α (εγ + εω )
(2.117)
Exponentially decaying terms have all a similar form which is a sum of tn e−kαt where
n and k are integers. Hence, the time for the relaxation of the centered moments
tends toward infinity when α tends toward zero. This represents our first constraint,
we need t α1 , that is, in terms of orders of magnitude, under the condition that the
volume fractions are rather similar (same order of magnitude)
t
1
h
(2.118)
Going back to the discussion about the two separated tubes, we now see why such
a system can not be described by a single equation since h = 0 in this case and the
relaxation time tends toward infinity. In addition, it is important to notice that, at
this point, we have reached the same conclusions that in [295] and found a similar
expression for D∞ , without proving it from the study of the higher order moments.
101
102
2.2.3 Centered moments for the weighted average
To make further progress, considering our interest in the weighted concentration
defined by Eq. 1.16, we study the weighted central moments defined by Eq (2.119)
mγω
n =
εγ
εω
mγn +
mω
εγ + εω
εγ + εω n
(2.119)
The different values for these moments are presented below. It is important to
keep in mind that the centered moments, in the long time limit, for the one-equation
models are those of a normal distribution, that is,
n
mγω
n
= (n − 1)!! (2D∞ t) 2 for n even
Qγω
mγω
n
= 0 for n odd
Qγω
FI R S T O R D E R
mγω
1
= V ∞ t + exp
γω
Q
(2.120)
There are two important conclusions that can be formulated from Eq (2.120).
• In this expression, it is apparent that the first order weighted centered moments
is propagating at V ∞ at long times and that the relaxation is purely driven by
the exponential terms. However, even though the constant shift disappears
when considering the weighted concentration, it is still unclear why the signal
can be described by a Gaussian as, a priori, the constant shift still induces a
strong effect on the higher order moments.
• It also shows that the relaxation toward a one-equation model is faster for the
weighted average concentration than for the intrinsic concentrations. There
has been some discussions about what should be used as a macroscopic concentration for a one-equation model in a multiphase configuration [77]. For
example, in a porous medium colonized by biofilms, experimenters tend to use
the concentration only in the water-phase whereas the results herein suggest
that the weighted concentration should be considered, and this would bring
some simplified behavior.
At this point, we make the following approximations Vγω − Vωγ ∆V and Dγω −
Dωγ ∆D. These are not necessary but (1) they lead to interesting simplifications
and (2) they are verified in most cases, as, Vγγ and Vωω contain the dominant terms
hvγ iγ and hvω iω , unlike Vγω and Vωγ . Hence, ∆V and ∆D can be interpreted as the
velocity and dispersion contrasts between both phases.
Solving the differential equations leads to Eq (2.121)
SECOND ORDER
γω
3 εγ εω
1 m2
=−
Ω2 + D∞ t + exp
γω
2Q
2 (εγ + εω )2
(2.121)
This expression shows that a constant is still involved. Hence, an obvious constraint
for the asymptotic regime to be reached is Eq (2.122).
t
3 εγ εω
2
2 (εγ +εω )2 Ω
∞
(2.122)
D
This represents a sufficient constraint for the convergence of the second order
centered moment but it is still unclear what is happening for higher orders moments.
Considering the third order moment leads to Eq (2.123)
THIRD ORDER
γω
1 m3
2 Qγω
=
9
εγ εω
α2 (εγ + εω )2
+ 6
∆V∆D + 8
εγ εω
α (εγ + εω )
εγ εω (εγ − εω )
α3 (εγ + εω )3
∆V∆D + 3
2
∆V 3
!
∆V 3 t + exp
α2 (εγ + εω )3
(2.123)
For a Gaussian shaped signal, the third order centered moment is zero, like other
odd order moments. In our case, it tends toward infinity as a O (t). At first, this is quite
disheartening and we clearly do not have a convergence in terms of the centered
moments. Thus, it is legitimate to wonder how the description of this signal can
be undertaken using a one-equation model. In the next section, we show that the
O (t) behavior can be neglected as compared to the weighted spreading of the signal,
γω
measured by m2 .
FOURTH ORDER
γω
1 m4
= c4 + C4 t + 4 (D∞ t)2 + exp
γω
3Q
(2.124)
We see that the dominant term at long time in Eq (2.124) is (D∞ t) which is the
fourth order moments corresponding to Eq (2.78).
2
N-TH ORDER
We were unable to formally prove a general expression for the n-th
order centered moment. However, the calculation of these, up to the sixth order, leads
to the following conclusions
∞ 2
n
2
• We have m
for n even, with the double factorial
Qγω = (n − 1)!! (2D t) + O t
Q
(n − 1 − 2i).
given by (n − 1)!! =
γω
n
i;062in−1
n
103
104
mγω
n
−1 for n − 1 odd.
• We have Qn−1
γω = O t 2
2.2.4 Standardized moments, skewness and kurtosis
The goal of this section is to understand the time-asymptotic behavior of the standardized moments defined by, for n > 2,
Mγω
n
mγω
n
=
n
γω 2
m2
(2.125)
For the one-equation model, with a Dirac input, we have, at long times
Y
Mγω
n = (n − 1)!! =
(n − 1 − 2i) for n even
i;062in−1
Mγω
n
= 0 for n odd
In our case, we can calculate the time-infinite limit of the standardized moments
on the basis of the equations given in section 2.2.3. For the orders 3 and 4, this leads
to:
SKEWNESS
lim Mγω
3 =0
t→∞
(2.126)
KURTOSIS
lim Mγω
4 =3
t→∞
(2.127)
In the context of statistical physics, the skewness and the kurtosis are often used to
test for the normality of a set of data, that is, these are considered sufficient measures
for the normality of a distribution. In our case, it seems that the signal tends toward a
Gaussian in the sense of the skewness and kurtosis. In a more rigorous perspective, it
is necessary to consider the general formulation of the leading terms of all the higher
order centered moments. From the results extrapolated in section 2.2.3, we have
n
n mγω
mγω
n
n−1
∞ n
2
2
(n
(2
=
−
1)!!
D
t)
+
o
t
,
n
even
and
=
O
t 2 −1 , n odd
Qγω
Qγω
It is straightforward that for n even,
lim Mγω
n = (n − 1)!!
t→∞
and for n odd,
lim Mγω
n =0
t→∞
that is, there is a convergence of the two-equation model toward a one-equation
model in terms of standardized moments. In other words, it means that we never
rigorously, i.e., for finite t, have a normal distribution of the concentrations but that
the spreading, i.e., the second order centered moment, is dominating all other kind
of deformation of the signal at long times.
2.2.5 Constraints and convergence
In this section, we develop the constraints that indicate when the one-equation model
can be used instead of the two-equation one. First of all, as previously discussed, we
need
Constraint A: t 1
h
(2.128)
In addition, for the second order, we need constraints such as Eq (2.129).
3 εγ εω
Ω 2 D∞ t
2
2 (εγ + εω )
(2.129)
To develop clear constraints that apply to any order, we define a measure of this difference in terms of standardized moments. We say that the one-equation formulation
represents a good approximation of the transport processes if and only if
1. Constraint A: t h1 is satisfied
2. Constraint B: For n > 2, we have δn = |
∞
mγω
n −mn
(
m∞
2
)
n
2
| 1 where m∞
n is the nth-
order moment of the Gaussian asymptotic one-equation model.
SECOND ORDER
With this definition, we still have the same constraint associated
with the second order Eq (2.130).
3 εγ εω Ω2
t
2 (εγ + εω )2 D∞
(2.130)
There is an interesting physical interpretation for this constraint. Going back to Fig
(27), we see that it means that the distance between the peaks of the signals in each
phase must be small as compared to the spreading of the signals.
105
106
In addition, the definition of D∞ is
D∞ = εγ (Dγγ + Dγωε) ++εεω (Dωγ + Dωω) +
γ
ω
εγ εω
α (εγ + εω )
2
∆V 2
(2.131)
and obviously
D∞ >
εγ εω
α (εγ + εω )
2
∆V 2
(2.132)
because
εγ (Dγγ + Dγω ) + εω (Dωγ + Dωω )
>0
εγ + εω
(2.133)
Going back to the definition of Ω, we have
Ω=
1
∆V
α
(2.134)
Hence, a sufficient condition for the expression of this constraint on the second
order term is also
t
1
α
(2.135)
Obviously, if the volumic fraction occupied by one of the phases is negligible, then
the two-equation model can be replaced by a one-equation description. This is an
important result, but we will not consider such simple cases. We assume that the
volumic fractions of each phase are rather similar (not separated by several orders of
magnitudes), so that the sufficient condition boils down to
t
1
h
(2.136)
with a convergence in O 1t .
THIRD ORDER
For the third order moment, Constraint B can be expressed by Eq
(2.137)
9
εγ εω
h2 (εγ + εω )
∆V∆D+8
2
!
3
εγ εω
3 +2 6
3 t (2D∞ t) 2
∆V
∆V∆
D
+
3
∆V
h3 (εγ + εω )3
h (εγ + εω )2
h2 (εγ + εω )3
(2.137)
107
As a first step toward the derivation of a sufficient condition associated with Eq
(2.137), we impose the inequality Eq (2.138):
1
h
9
εγ εω
h (εγ + εω )
∆V∆D + 8
2
∆V 3
h2 (εγ + εω )3
!
εγ εω
2 6
h (εγ + εω )
∆V∆D + 3
2
(2.138)
In terms of orders of magnitude, a sufficient condition for Eq (2.138) to be true is
Constraint A: t 1
h
As a consequence of imposing Eq (2.138), the inequality described by Eq (2.137)
now reduces to Eq (2.139)
2 6
εγ εω
h (εγ + εω )
∆V∆D + 3
2
h2 (εγ + εω )3
!
∆V 3 t (2D∞ t) 2
3
(2.139)
This can be simplified, and leads to an equivalent inequality Eq (2.140)
1
2
6
∆V∆D
εγ εω
h (εγ + εω ) (D∞ )
2
3
2
+3
!2
∆V 3
h2 (εγ + εω )3 (D∞ ) 2
3
!
∆V 3 t
h2 (εγ + εω )3
t
(2.140)
A sufficient condition, in terms of orders of magnitude, is Eq (2.141) in conjunction
with Eq (2.142)
1 ∆V 2 ∆D2
t
h 2 ( D∞ ) 3
(2.141)
1 ∆V 6
t
h4 (D∞ )3
(2.142)
Using the definition of D∞ Eq (2.131), it is straightforward to show that
1 ∆V 2 ∆D2
61
h ( D∞ ) 3
(2.143)
1 ∆V 6
61
h3 (D∞ )3
(2.144)
and
108
Figure 31: Convergence of the two-equation quasi-stationary model toward the one-equation
asymptotic model in terms of standardized moments measure δn for Vωγ =
Vγω = Dωγ = Dγω = 0, Vγγ = 5, Vωω = 1, Dγγ = 10, Dωω = 5 and h = 1.
and hence,
Constraint A: t 1
h
is a sufficient condition, for a convergence in O
√1
t
.
The convergence is polynomial in terms of standardized moments, in opposition
to the classical view of the single-continuum problem which is an exponential convergence in terms of centered moments [33]. For theskewness,
the expression of the
error given by Constraint B tends toward zero as a O √1
t
1
t
whereas, for the kurtosis, it
tends toward zero as a O
. This difference in the convergence means that the signal
recovers quicker the “peakedness” of the normal distribution than the symmetry.
P
Notice that δn can also be expressed as series such as n an 1n , leading to similar
conclusions regarding the long-time convergence.
t2
2.2.6 Numerical simulations
In this section, we are interested in solving Eqs (2.82) and (2.83) numerically and then
calculating the expression of the δn . Our goal is to sudy numerically the behavior of
the moments, up to the order six. We used the MATLABT M solver ode15s with a relative
tolerance of 2.22045 10−14 to provide some direct evidences for the Constraint: t 1
h . A convergence study concerning the tolerance parameter was performed showing
that, for the chosen one, no significant discretization error was apparent. Results
are presented Fig (31) up to the sixth order for δn . The results are given on a log-log
graph, highlighting two different regimes. For small times, the dominant term is the
Fully
non-local
&
Two-equation
time non-local
Two-equation
quasi-steady
One-equation
time asymptotic
2
Figure 32: Schematic summary of the domains of validity of the different models for a pulse
in a dual-phase/region infinite porous medium. τ1 is a characteristic time for the
relaxation of the effective parameters and τ2 is a characteristic time associated
with the mass exchange coefficient in the two-equation model.
exponential part. For longer times, the various errors tend linearly
toward zero with
a − 21 or −1 slopes, that is, the convergence is driven by a O √1
t
or O 1t and this
is coherent with the theoretical analysis. Direct numerical resolutions of the closure
problems for the different macroscopic models, comparisons with Direct Numerical
Simulations (DNS) and associated breakthrough curves can be found in [78] on a
simple 2-D geometry.
2.2.7 Discussion
In essence, this part is an extension of [295] and [177, 178, 237, 205] in that we (1) provide a comprehensive framework that can be used by experimenters for the choice of
the dual-continua models that are used at the Darcy-scale to describe mass transport,
(2) solve the issue of higher order moments in the one-equation situation and (3)
study the convergence of the two-equation model toward the one-equation formulation. Assuming that the spatial non-locality can be separated from the temporal
non-locality and that the hypothesis of separation of length scales is valid can lead to
various Darcy-scale models. These correspond to specific statements that are made
concerning the characteristic times of the problem. For the case of a pulse propagating through an infinite dual-phase/region porous medium, the different constraints
are summarized Fig (32), in terms of orders of magnitudes. Experimenters should
consider carefully these different limitations and mathematical models should be
chosen on the basis of solid physical reasons.
109
110
2.3
P E C U L I A R P E R T U R B AT I O N D E C O M P O S I T I O N
( MODEL
E)
A different way to obtain one-equation local non-equilibrium models consists in
decomposing concentrations as follows
ci = hciγω + ĉi
(2.145)
In this case, the fluctuation is defined in relation to the weighted average on both
phases/regions. The problem is closed using [205] Eq (52) (in which η is used instead
of γ), [178] Eqs (42) and (43), for heat transfer. It has been proposed in [205] for the
large-scale averaging problem and applied in [77] for a reactive case (linear kinetics).
The closure can be written
ĉi = bi · ∇ hciγω
(2.146)
This decomposition may be very efficient since it allows to directly develop a
macroscopic one-equation model, to reduce the number of closure parameters and
it is also useful for upscaling various class of problems such as reactive ones. The
counterpart is that the assumptions behind the first order closure and the quasistationarity of the fluctuations problems are stronger in this case because (1) the
closure recovers less characteristic times than the c̃ decomposition and (2) the norm
of the cî perturbation is larger than the norm of c̃i . The macroscopic mass balanced
equation takes the form [205]Eq (56) (in which η is used instead of γ)
(εγ + εω ) ∂t hciγω + V · ∇ hciγω = ∇ · {D∗ · ∇ hciγω }
(2.147)
The ĉi perturbation decomposition leads to the following expression of dispersion,
see in [205]Eq (57) (in which η is used instead of γ) and [78].
D∗ =
X
i=γ,ω
h
i
εi Di · I + h∇bi ii − hvi bi ii
(2.148)
The determination of effective parameters requires the calculation of a single problem which can be found in [205] Eqs (53) to (55) (in which η
is used instead of γ) and in [78].
CLOSURE PROBLEM
∗
vγ · ∇bγ = ∇ · {Dγ · ∇bγ } − ṽγ − ε−1
γ U
(2.149)
2.4 L O C A L M A S S E Q U I L I B R I U M ( MODEL F )
−nγσ · (
B.C.1 :
B.C.1 :
Dγ · ∇bγ) = nγσ · Dγ
bγ = bω
B.C.2 : −nγω · jbγ − jbω = −nγω · (Dω − Dγ )
−nωσ · (
B.C.4 :
Dω · ∇bω) = nωσ · Dω
bi (x + pk ) = bi (x)
Periodicity :
ii
hbi = 0
Unicity :
∗
vω · ∇bω = ∇ · {Dω · ∇bω } − ṽω + ε−1
ω U
on S γσ
(2.150a)
on S γω
(2.150b)
on S γω
(2.150c)
on S ωσ
(2.150d)
k = 1, 2, 3 (2.150e)
(2.150f)
(2.151)
with
U∗ = εγ εω
2.4
hvγ iγ − hvω iω
εγ + εω
LOCAL MASS EQUILIBRIUM
(2.152)
( MODEL
F)
The local mass equilibrium situation corresponds to
∼ hcω iω =
∼ hciγω
hcγ iγ =
(2.153)
In other words, this means that the concentration gradients within the phases/regions are sufficiently small to extend the thermodynamic equilibrium at the interface
to the bulk phases/regions. This represents a very particular physical situation for
which the total mass within the medium can be described by a single weighted conγω
centration hci , that is, by a one-equation model such as Eq (2.154). The associated
constraints have been extensively discussed and can be found in [274, 114, 287, 260].
These are usually expressed in terms of orders of magnitude of dimensionless numbers. In this situation, a reasonable approximation [274] of the two-equation model
is [205] Eq (20) (in which η is used instead of γ)
(εγ + εω ) ∂t hciγω + V · ∇ hciγω = ∇ · {Dequ · ∇ hciγω }
(2.154)
It can be obtained by summing Eqs (2.38) and (2.39) and, therefore, the local mass
equilibrium dispersion tensor can be written [205] Eq (22) (in which η is used instead
of γ)
Dequ =
X
i,j=γ or ω
Dij =
X
i=γ,ω
εi hDi · (I + ∇Bi ) − ṽi Bi ii
(2.155)
111
112
with a closure on the perturbations that takes the form
c̃i = Bi · ∇ hciγω
The effective velocity takes the very simple form [205]Eq (23) (in which η is used
instead of γ)
γ
V = εγ hvγ i + εω hvω i
ω
(2.156)
The closure parameters Bi are simply related to those of the two-equation model
by
Bγ = bγγ + bγω
(2.157a)
Bω = bωγ + bωω
(2.157b)
or they can be calculated solving directly the following boundary value problem at
the microscale
∗
vγ · ∇Bγ = ∇ · {Dγ · ∇Bγ } − ṽγ − ε−1
γ β
B.C.1 :
B.C.1 :
−nγσ · (
(2.158)
Dγ · ∇Bγ) = nγσ · Dγ
Bγ = Bω
B.C.2 : −nγω · jBγ − jBω = −nγω · (Dω − Dγ )
B.C.4 :
−nωσ · ( ω · ∇Bω ) = nωσ · ω
D
Periodicity :
D
Bi (x + pk ) = Bi (x)
hBi ii = 0
Unicity :
∗
vω · ∇Bω = ∇ · {Dω · ∇Bω } − ṽω + ε−1
ω β
with
1
β =−
V
on S γσ
(2.159a)
on S γω
(2.159b)
on S γω
(2.159c)
on S ωσ
(2.159d)
k = 1, 2, 3 (2.159e)
(2.159f)
(2.160)
Z
∗
Aγω
nγω · jBγ dA
(2.161)
3
F O R M A L E Q U I VA L E N C E B E T W E E N MODEL D AND E
The goal of this part is to obtain a formal proof of the equivalence between Eqs
(2.77) and (2.147), that is, to prove D∗ = D∞ .
3.1
REMINDER OF THE EXPRESSIONS FOR THE DISPERSION TENSORS
3.1.1 One-equation time-asymptotic non-equilibrium model
The dispersion tensor Eq (1.25) can be written [295] Eqs (97) to (99)
X
D∞ =
Dij +
i,j=γ or ω
= Dequ +
1
(U∗ + dω − dγ ) (U∗ − β∗ )
h
1
(U∗ + dω − dγ ) (U∗ − β∗ )
h
Both parts of this expression have very distinct physical meaning.
(3.1)
P
i,j=γ or ω
D∗∗
ij repre-
sents the sum of the dispersion terms, which strictly speaking, corresponds to local
mass equilibrium dispersion, and is rather similar to the single phase situation. However, the second part stands for the multiphase aspects. It expresses the contrast effect
between the γ − phase and the ω − phase. It tends toward 0 when the exchange
coefficient α∗∗ tends toward infinity and is mainly driven by the square of
U
∗
hvγ iγ − hvω iω
= εγ εω
εγ + εω
(3.2)
It also contains
∗
β
dω − dγ
Z
1
nγω · jBγ dA
= −
V Aγω
X
=
εi hDi · ∇ri − ṽi ri ik
(3.3a)
(3.3b)
i=γ,ω
This asymptotic behavior was proved directly from the lower-scale equation in
[166] for stratified systems, and used extensively in [150] which introduced the word
113
114
F O R M A L E Q U I VA L E N C E B E T W E E N
model d and e
Taylor’s dispersion, for that particular case, because of the square dependence of the
dispersion coefficient with the velocity difference.
Notice that there are usually six closure (or mapping) variables associated to the
two-equation model, that is bγγ , bγω , bωγ , bωω , rγ , rω . In the expression of the dispersion Eq (3.1) only four closure variables remain, namely rγ , rω and Bγ , Bω . They
are simply related to those of the two-equation model by
Bγ = bγγ + bγω
Bω = bωγ + bωω
(3.4a)
(3.4b)
CLOSURE PROBLEMS
The set of effective parameters previously introduced can
be computed through the resolution of the two following closure problems. They can
be straightforwardly derived from [2] Eqs (49) to (56) (in which η is used instead of γ).
vγ · ∇rγ = ∇ · {Dγ · ∇rγ } − ε−1
γ h
B.C.1 :
B.C.1 :
−nγσ · (
(3.5)
Dγ · ∇rγ) = 0
rγ = rω
B.C.2 : −nγω · jrγ − jrω = 0
D
on S γσ
(3.6a)
on S γω
(3.6b)
on S γω
(3.6c)
B.C.4 : −nωσ · ( ω · ∇rω ) = 0 on S ωσ
Periodicity :
ri (x + pk ) = ri (x)
k = 1, 2, 3
Unicity :
hri ii = 0
(3.6d)
(3.6e)
(3.6f)
vω · ∇rω = ∇ · {Dω · ∇rω } + ε−1
ω h
(3.7)
∗
vγ · ∇Bγ = ∇ · {Dγ · ∇Bγ } − ṽγ − ε−1
γ β
(3.8)
3.1 R E M I N D E R O F T H E E X P R E S S I O N S F O R T H E D I S P E R S I O N T E N S O R S
B.C.1 :
B.C.1 :
−nγσ · (
Dγ · ∇Bγ) = nγσ · Dγ
Bγ = Bω
B.C.2 : −nγω · jBγ − jBω = −nγω · (Dω − Dγ )
B.C.4 :
−nωσ · ( ω · ∇Bω ) = nωσ · ω
D
Periodicity :
D
Bi (x + pk ) = Bi (x)
on S γσ
(3.9a)
on S γω
(3.9b)
on S γω
(3.9c)
on S ωσ
(3.9d)
k = 1, 2, 3
(3.9e)
hBi ii = 0
Unicity :
(3.9f)
∗
vω · ∇Bω = ∇ · {Dω · ∇Bω } − ṽω + ε−1
ω β
(3.10)
3.1.2 One-equation special perturbation decomposition
The dispersion tensor takes the form Eq (3.11)
D∗ =
X
i=γ,ω
h
i
εi Di · I + h∇bi ii − hvi bi ii
(3.11)
CLOSURE PROBLEM
The determination of effective parameters requires the calculation of a single problem which can be found in [205] Eqs (53) to (55) (in which η
is used instead of γ) and in [78].
∗
vγ · ∇bγ = ∇ · {Dγ · ∇bγ } − ṽγ − ε−1
γ U
B.C.1 :
B.C.1 :
−nγσ · (
(3.12)
Dγ · ∇bγ) = nγσ · Dγ
bγ = bω
B.C.2 : −nγω · jbγ − jbω = −nγω · (Dω − Dγ )
B.C.4 :
−nωσ · ( ω · ∇bω ) = nωσ · ω
D
Periodicity :
Unicity :
D
bi (x + pk ) = bi (x)
hbi ii = 0
∗
vω · ∇bω = ∇ · {Dω · ∇bω } − ṽω + ε−1
ω U
on S γσ
(3.13a)
on S γω
(3.13b)
on S γω
(3.13c)
on S ωσ
(3.13d)
k = 1, 2, 3
(3.13e)
(3.13f)
(3.14)
115
116
3.2
model d and e
M AT H E M AT I C A L D E V E L O P M E N T
At this point, the mapping variables rγ , rω , Bγ , Bω and bγ , bω are solutions of different
boundary value problems Eqs (3.5) to (3.7), Eqs (3.8) to (3.10) and Eqs (3.12) to (3.14).
In order to find a relationship between these closure variables, we introduce the
following source terms decomposition based on the superposition principle for linear
operators
Bi = BIi + BIIi β∗
(3.15a)
∗
(3.15b)
bi = bIi + bIIi U
ri = rIIi α∗∗
(3.15c)
As a consequence of the previous decomposition, Eqs (3.5) to (3.7), Eqs (3.8) to
(3.10) and Eqs (3.12) to (3.14) reduce to only two boundary value problems. BIi and
bIi are solutions of the following closure problem Type I
vγ · ∇ΦIγ = ∇ · {Dγ · ∇ΦIγ } − ṽγ
B.C.1 :
−nγσ · (
Dγ · ∇ΦIγ) = nγσ · Dγ
ΦIγ = ΦIω
B.C.2 : −nγω · jΦIγ − jΦIω = −nγω · (Dω − Dγ )
B.C.1 :
B.C.4 :
Periodicity :
−nωσ · (
Dω · ∇ΦIω) = nωσ · Dω
ΦIi (x + pk ) = ΦIi (x)
(3.16)
on S γσ (3.17a)
on S γω (3.17b)
on S γω (3.17c)
on S ωσ (3.17d)
k = 1, 2, 3 (3.17e)
vω · ∇ΦIω = ∇ · {Dω · ∇ΦIω } − ṽω
(3.18)
Unicity of the solutions is provided by the following conditions
εγ hBIγ iγ + εω hBIω iω = 0
γ
εγ hbIγ i + εω hbIω i
ω
= 0
(3.19a)
(3.19b)
BIIi , bIIi and rIIi are solutions of the following closure problem Type II
vγ · ∇ΦIIγ = ∇ · {Dγ · ∇ΦIIγ } − ε−1
γ
(3.20)
3.2 M AT H E M AT I C A L D E V E L O P M E N T
B.C.1 :
B.C.1 :
−nγσ · (
Dγ · ∇ΦIIγ) = 0
ΦIIγ = ΦIIω
B.C.2 : −nγω · jΦIIγ − jΦIIω = 0
B.C.4 : −nωσ · ( ω · ∇ΦIIω ) = 0
D
Periodicity :
ΦIIi (x + pk ) = ΦIIi (x)
on S γσ
(3.21a)
on S γω
(3.21b)
on S γω
(3.21c)
on S ωσ
(3.21d)
k = 1, 2, 3
(3.21e)
vω · ∇ΦIIω = ∇ · {Dω · ∇ΦIIω } + ε−1
ω
(3.22)
Unicity of the solutions is provided by the following conditions
εγ hBIIγ iγ + εω hBIIω iω = 0
εγ hbIIγ iγ + εω hbIIω iω = 0
1
hrIIω iω = ; hrIIγ iγ = 0
h
(3.23a)
(3.23b)
(3.23c)
bIi and BIi are solutions of the same problem and satisfy the same unicity equation.
Hence, we have
bIi = BIi
(3.24)
which implies
bi = Bi − BIIi β∗ + bIIi U∗
(3.25)
It is also important to notice that bIIi and BIIi are solution of the same problem Type
II and satisfy the same unicity equations so that we have
bi = Bi + bIIi (U∗ − β∗ )
(3.26)
Finally bIIi and rIIi are solution of the same problem but do not satisfy the same
unicity equations so that they only differ by a constant. After some simple algebra, we
obtain
1
εω
(U∗ − β∗ )
bi = Bi + ∗∗ ri −
α
(εγ + εω )
(3.27)
117
118
model d and e
Table 5: Equivalence of the approximations associated to the volume averaging theory and to
the moments matching technique.
3.3
Approximation
Moments matching
Volume averaging
of
technique
theory
Convolutions
Spatial moments matching
First order
in space
up to the second order
closure
Convolutions
Time-infinite behavior
Quasi-stationarity
in time
of the spatial moments
of the closure problems
DISCUSSION
Injecting this expression of bi , i.e. Eq (3.27), in Eq (3.11) straightforwardly leads to
D∗ = D∞
(3.28)
One may realize that the physics underlying the time-asymptotic hypothesis in
one-phase or two-phase systems have very different backgrounds. In the single-phase
configuration, the equality of the dispersion tensors between the volume averaging
theory as devised in [274] and the moments matching technique as devised in [33]
is straightforward. In this single-phase case, the problem is not homogenizable at
very short times, that is, one needs to consider only long times to avoid the convolutions. This limitation corresponds to the time needed for a particle to visit the entire
microscopic domain. In a dual-phase situation, this assumption of the single-phase
case is reminiscent to the quasi-stationary hypothesis on the standard perturbations, although not exactly similar because of the contrast of properties and of the
exchange between both phases. The time-asymptotic behavior of the two-equation
model corresponds to a macroscopic relaxation of the two-equation boundary-valueproblem, that is, a relaxation of the first two spatial moments or of the nonstandard
perturbation. This represents a very different approximation.
The result expressed by Eq (3.28) provides a direct equivalence between the moments matching method and a special volume averaging theory based on a different
decomposition technique. This conclusion has various consequences
• Two-equation models provide a solid basis for exploring physical aspects of the
dispersion problem.
• Going back to the discussion about the choice of the proper mass transfer
coefficient (see [151, 186]) the results obtained in this paper show that the other
proposed values would lead to an incorrect asymptotic dispersion equation,
even if they can work better at some limited stages of the transient evolution.
3.3 D I S C U S S I O N
• We use the moments matching technique on an already homogenized set of
equations, that is, we do not start at the microscopic scale as proposed in [33].
Although it has not been devised yet, we believe that the two-equation model
can be obtained using the method in [33] and the equivalence developed in
this article strongly reinforces the relationship between the volume averaging
theory and the moments matching techniques.
• This equality Eq (3.28) also allows to see a similar problem from two different
viewpoints Table 5. The long time limit is traduced, in the ĉ decomposition, as
a quasi-stationary closure problem and this is a new interesting way of seeing
the time-asymptotic hypothesis in the multiphase configuration. Moreover,
Eq (3.28) shows that a very strong relationship exists between the closure on ĉ
and the moments matching method limited to the second order, as applied in
[295, 2].
119
4
N U M E R I C A L S I M U L AT I O N S O N A S I M P L E E X A M P L E A N D
D O M A I N S O F VA L I D I T Y
In this part, we are interested in computing some of the previous models on a simple
2D geometry (Fig. 33) in order to catch the main characteristics of the problem and to
delineate their domain of validity. In particular, we study the response of models local
in times because (1) these are the most suited, because of their inherent simplicity, for
experimental studies and (2) developing a computational method for the resolution
of the complex integro-differential equations (with the convolutions) is beyond the
scope of this thesis. We show that even for an input signal introducing many characteristic times, the time-asymptotic model may give a rather good approximation at
short times and gains in precision as the time tends toward infinity.
The γ − phase is convective and diffusive whereas the ω − phase is only diffusive
(called Mobile-Immobile situation). One example of such a system could be the study
of a tracer transport between two plates colonized by biofilms (aggregations of microorganisms coated in protective extra-cellular substances). The closure problems Type
I and Type II (Fig. 34) presented in the next section are solved using the ComsolT M
multiphysics package. Notice that because of the very particular geometry and because we impose periodic boundary conditions, the closure field is 1D. We also fully
solve the balanced momentum equations so that we do not consider any upscaling
Figure 33: Schematic description of the 2D system
121
122
N U M E R I C A L S I M U L AT I O N S O N A S I M P L E E X A M P L E A N D D O M A I N S O F VA L I D I T Y
Figure 34: Norm of the bi field for P e = 200
of momentum equations. The closure problems only depends upon a Péclet number
defined as
Pe=
hvγ iγ L
Dγ
(4.1)
where we choose
L = 1
Dω
= 0.8
Dγ
εω = 0.2
(4.2a)
(4.2b)
(4.2c)
The dimensionless time is defined by
t0 =
hvγ i t
L
(4.3)
and the concentration is normalized to the amplitude of the input concentration.
On the one hand, we solve the entire 2D microscopic problem on a total length
of 60L (called DNS for Direct Numerical Simulation) for a square input for different
Péclet numbers. On the other hand, we solve the 1D upscaled local equilibrium,
non-equilibrium and two-equation models on a total length of 60L (Fig. 35). Then,
we observe breakthrough curves at 10L and 50L for Péclet numbers of 2, 20 and 200.
Pe=2
P e = 20
P e = 200
Figure 35: Breakthrough curves for various P e after a) 10L and b) 50L for a square input of
width δt 0 = 5 starting at t 0 = 0. The solid line corresponds to the DNS, the dotted
line to the local mass equilibrium model, the dashed line with triangles to the
two-equation model and the dashed-dotted line with stars to the time-asymptotic
model.
On Fig. 35, we see that the three homogenized models provide a very good approximation of the transport problem. At low Péclet, time and space non-locality
tend to disappear because time and length scales are separated by several orders of
magnitude. Meanwhile, some very little discrepancy still remains at the peak of the
signal at 10L. At the very beginning of the system, the time-width of the signal propagating is of the same order of magnitude as the characteristic time for the relaxation
of the effective parameters. When the signal spreads, the non-locality disappears and
at 50L all the signals are in good agreement. The propagation is even slow enough
for the local mass equilibrium assumption to be well-founded, that is, the exchange
coefficient is big enough for the multiphase contrast term in the expression of the
time-asymptotic dispersion Eq (3.1) to be insignificant.
123
124
Figure 36: Domains of validity of the different models as a function of the Péclet number and
of the time for a square input signal.
When the Péclet number reaches values around 20, the local mass equilibrium
assumption starts to become inappropriate. Fig. 35 shows that the local mass equilibrium model gives a poor approximation of the signal whereas both non-equilibrium
models are still in good agreement. The fact that the peaks for these two models arise
earlier than the one of the DNS is again characteristic of non-locality. Memory functions or fully non-local theories (such as n-equations models) should be considered
in this case.
For Péclet numbers around 200, Fig. 35, the local mass equilibrium model is completely inaccurate. The two-equation model provides the best approximation because
it captures more characteristic times than the time-asymptotic one. Except for nonlocality (especially at 10L), it recovers the shape of the signal. However, even for
a square input signal and high Péclet numbers, the time-asymptotic model is still
rather accurate. As the signal spreads, all the non-equilibrium models tend toward
the correct solution and this means that domains of validity need a time dimension.
Results suggest that the one-equation local non-equilibrium model might represent,
in cases such as intermediate Péclet numbers, macroscopic stationarity or asymptotic regimes, a good compromise, in terms of computational demand, between the
two-equation and the local mass equilibrium models. The importance of non-locality
is also emphasized and becomes particularly obvious in the high Péclet number
situation.
On the basis of these results, domains of validity can be determined for a square
input signal identical in both phases Fig. 36. When the Péclet number is below or
approximately unity, the local mass equilibrium condition is verified except at very
short times where fully non-local theories or n-equation models should be considered.
Above unity, the situation is more complex as three different regimes are identified.
At very short times, the situation can not be described even by the quasi-stationary
two-equation model and non-local theories are necessary. At intermediate times, the
two-equation model represents the only alternative to convolutions. As the time tends
toward infinity and the signal spreads, the one-equation non-equilibrium model can
be used to describe the mass transport. The boundaries between these different
regimes depend, among others, on the input boundary condition, on the microscopic
topology and on the processes. The constraints associated to the boundary (1) and
(2), Fig. 36, between the non-local and local zone have been extensively discussed
[1, 2, 48, 113, 43]. However, little is known on the limitations associated with (3), Fig.
36, and it requires further investigation.
Concerning the influence of the boundary conditions on the domains of validity,
it is important to emphasize that two-equation models allow to modify separately
the conditions for each phase unlike one-equation models. As a consequence, in
situations where the boundary conditions in one phase are very different from the conditions in the other phase, two-equation models must be considered at the expense
of one-equation models. Additionally, two-equation models are likely to provide a
better approximation of the transport processes when many modes are excited, that
is, for a Dirac input for example.
125
5
CONCLUSION
This part presents a comprehensive macrostransport theory in dual-phase and dualregion porous media. In particular, we show the following fundamental points
• The method of volume averaging represents an adapted framework for this
purpose and allows to develop various deterministic models, capturing more
or less information from the microscale processes, that can be used to describe
the Darcy-scale transport with biofilms, or larger scale heterogeneities. The
domains of validity are well defined on the basis of theoretical analyses (in
particular spatial moments matching) and numerical simulations.
• The fundamental analysis carried out in section 3 leads to D∞ = D∗ . It has
broad practical implications since these one-equation models, because of their
intrinsic simplicity, are widely used by experimenters. In addition, these models
were obtained using very distinct techniques, that is, moments matching for the
time-asymptotic model and a closure on a peculiar perturbation for the model.
From a theoretical point of view, equivalence between these two methods
shows that (1) two-equation quasi-steady models provide a reliable basis for
the study of the dispersion problem, (2) the idea of matching moments up to
the second order is similar to the closure on the spatial perturbations and (3)
the time-asymptotic limit of the moments corresponds to the quasi-stationarity
of the perturbation problem.
• The numerical results obtained for a Mobile-Immobile problem with different
Péclet numbers show that (1) the local mass equilibrium model has a restricted
area of validity and must be used very carefully (see discussion in [73]), (2) the
one-equation non-equilibrium model produces a reasonable approximation
of the transport at long times, even for a square input signal and high Péclet
numbers, (3) the two-equation model gives, in all cases, better results and (4)
domains of validity representations need a time dimension.
127
Part IV
M O D E L I N G B I O L O G I C A L LY R E A C T I V E
NON-EQUILIBRIUM MASS TRANSPORT IN POROUS
MEDIA WITH BIOFILMS
1
INTRODUCTION
1.1
ABSTRACT
In this part, we develop a one-equation non-equilibrium model to describe the
Darcy-scale transport of a solute undergoing biodegradation in porous media. Our
approach is based on the development of a macrotransport theory reminiscent to the
dual-phase situation studied in the previous part, but in a reactive case. Most of the
mathematical models that describe the macroscale transport in such systems have
been developed intuitively on the basis of simple conceptual schemes. There are two
problems with such an heuristic analysis. First, it is unclear how much information
these models are able to capture; that is, it is not clear what the model’s domain of
validity is. Second, there is no obvious connection between the macroscale effective parameters and the microscopic processes and parameters. As an alternative,
a number of upscaling techniques have been developed to derive the appropriate
macroscale equations that are used to describe mass transport and reactions in multiphase media. These approaches have been adapted to the problem of biodegradation
in porous media with biofilms, but most of the work has focused on systems that
are restricted to small concentration gradients at the microscale. This assumption,
referred to as the local mass equilibrium approximation, generally has constraints
that are overly restrictive. In this work, we devise a model that does not require the assumption of local mass equilibrium to be valid. In this approach, one instead requires
only that, at sufficiently long times, anomalous behaviors of the third and higher
spatial moments can be neglected; this, in turn, implies that the macroscopic model
is well represented by a convection-dispersion-reaction type equation. This strategy
is very much in the spirit of the developments for Taylor dispersion presented by Aris
(1956). On the basis of our numerical results, we carefully describe the domain of
validity of the model and show that the time-asymptotic constraint may be adhered
to even for systems that are not at local mass equilibrium.
131
132
INTRODUCTION
1.2
CONTEXT
Biodegradation in porous media has been the subject of extensive studies from
the environmental engineering point of view [172, 220, 277, 278, 294]. Reactions
are mediated by microorganisms (primarily bacteria, fungi, archaea, and protists,
although others may be present) aggregated and coated within an extracellular polymeric matrix; together, these which form are generically called biofilms. There has
been significant interest for their role in bioremedation of soils and subsurfaces
[235, 227, 276, 97, 46, 29, 30] and, more recently, for their application to supercritical
CO2 storage [174, 64]. Numerous models for describing the transport of solutes, such
as organic contaminants or injected nutrients, through geological formations as illustrated in Fig (37), have been developed. Reviews of these mathematical and physical
representations of biofilms processes can be found in [79] and [181].
1.2.1 One-equation local mass equilibrium model
In many applications, the macroscopic balance laws for mass transport in such hierarchical porous media with biofilms have been elaborated by inspection. For example,
the advection-dispersion-reaction type equation (1.1) is commonly considered to
describe the Darcy-scale transport of a contaminant/nutrient represented by a concentration hcγ iγ in the water γ − phase. Brackets notations are here as a reminder
that this concentration must be defined in some averaged sense.
∂hcγ iγ
+ hvγ iγ · ∇hcγ iγ = ∇ · ( · ∇hcγ iγ ) + R γ
∂t
D
In this expression, hvγ iγ is the groundwater velocity and
(1.1)
D is a dispersion tensor. The
hc iγ
reaction rate is usually assumed to have a Monod form R γ = −α hc iγγ +K , where α
γ
and K are parameters (discussed in §2.5). It is common to assume that the solute
transport can be uncoupled from the growth process [285, 198], that is, to consider
that the characteristic times for these two processes are separated by several orders of
magnitude. This Monod expression can be extended to include both electron acceptor
and electron donor concentrations or simplified to a linear reaction rate in the limit
hcγ iγ K .
If one started using Eq (1.1) as an empirical representation of the mass transport and reaction process, it would not be immediately obvious how the microscale
processes influence each of the macroscale parameters that appear in the balance.
To understand how information is passed through the scales of observation, it is
necessary to start by considering the microscale physics of the phenomena. At the
pore-scale, biofilms in porous media are usually represented by convective-diffusive
processes within the fluid γ − phase, and diffusive-reactive processes within the
1.2 C O N T E X T
Figure 37: Hierarchy of the main scales
biofilm ω − phase. This representation is built on three assumptions: 1) the biofilm
is thick enough to be treated as a continuum [287, 285], 2) the rate of reaction of planktonic cells (suspended in the water-phase) can be neglected compared to biofilms
species (fixed on a surface and embedded within extracellular polymeric substances),
and 3) the microscale channels that sometimes form within the biofilms are treated
as part of the continuous fluid-phase. The mass exchange process between the fluid
and biofilm phases is described by a continuity of the flux and of the concentrations
133
134
INTRODUCTION
at the interface, and a zero-flux condition is applied to the solid boundaries. Equation
(1.1) represents a one-equation approximation of all these processes at the Darcyscale, and such an approach is frequently used in the literature. Nevertheless, merely
one physical situation, referred to as the local mass equilibrium, has been clearly
identified to be properly described by only hcγ iγ . In this case, the averaged concentrations in both phases are equal (strictly speaking, linked by a thermodynamical
constant often close to unity). In other words, when the gradients of the pointwise
concentrations within each phase can be neglected, the continuity at the interface
between the biofilms and the water-phases can be extended to the bulk phases and
the modelization can be undertaken using a one-equation model.
1.2.2 Multiple-continua models
Along with the identification of these limitations, some studies have worked out
models that capture more physics of the reactive transport. The fluid-biofilm system
has some obvious similarities to mobile-immobile configurations, and one might
consider a multiple-continua description for biofilms in porous media under some
circumstances. Several models have been developed with an explicit representation of
the multiple-region aspects of the reactive transport. These include the microcolony
[175] and idealized biofilm [65, 221, 219, 19] models in which the porous medium is
decomposed into a solid impermeable grain, a diffusive-reactive biofilm, a diffusive
boundary layer and an advective-diffusive bulk water-phase. This representation
leads to two-equation models where each equation describes the behavior of the
averaged concentration on one single phase, and there is exchange between phases
sharing common boundaries. Such models are able to capture more complex dynamics than one-equation local mass equilibrium models. Unfortunately, there are still
two difficulties with such an representations
1 There is only an intuitive (rather than formal) relationship between
the problem at the microscale and the one at the Darcy-scale. Hence, (1) it is still
unclear when this model should be applied instead of the one-equation local mass
equilibrium model for example, and (2) the dependence of the effective parameters
(dispersion, effective velocities, mass exchange coefficients and effective reaction
rate) on the microscale processes and geometry remains unknown.
PROBLEM
PROBLEM 2
The system of differential equations that need to be solved is more
complex, and, thus, is more difficult to use in applications.
As a solution to the Problem 1, one can find a more precise connection between the
macroscopic model and the associated microscale boundary value problem through
upscaling methods. The physics of non-reactive transport has been widely addressed
by deterministic techniques such as homogenization, moments matching and volume
averaging with closure. Such approaches have been adapted to the problem of reactive
1.2 C O N T E X T
transport with biofilms in porous media. These include the work of Wood et al. [284]
and Golfier et al. [114] who used the volume averaging with closure theory [274] to
compute effective parameters of the medium. However, except for two limit cases that
have been studied by Orgogozo et al. [191], most of the work with volume averaging
has focused on the local mass equilibrium assumption which is often excessively
restrictive. To address the non-equilibrium situation in this two-phase configuration,
one could consider two-equation models Eqs (1.2)
εγ
εω
∂ hcγ iγ
γ
ω
∗∗
+ V∗∗
γγ · ∇ hcγ i + Vγω · ∇ hcω i =
∂t
∂ hcω iω
∂t
γ
ω
∇ · D∗∗
+ ∇ · D∗∗
γγ · ∇ hcγ i
γω · ∇ hcω i
−h∗∗ (hcγ iγ − hcω iω )
(1.2a)
γ
ω
γ
ω
∗∗
∗∗
+ V∗∗
+ ∇ · (D∗∗
ωγ · ∇ hcγ i + Vωω · ∇ hcω i = ∇ · Dωγ · ∇ hcγ i
ωω · ∇ hcω i )
−h∗∗ (hcω iω − hcγ iγ ) + R ω
(1.2b)
∗∗
Here, hci ii is the concentration of the solute in the i−phase. V∗∗
ij and Dij (i and j are
dummy indexes for γ, water-phase, or ω, biofilm-phase) are the macroscopic velocities and dispersion tensors of the two-equation model and h∗∗ is the mass exchange
coefficient. Eqs (1.2) can be seen as a general compact way to write dual continua
models [58, 48, 47, 2, 115, 7, 81]. The previous denominations refer to the scale of
application and the physical processes involved whereas the two-equation models
definition refers to the mathematical structure of the problem. These have been
extensively used in hydrology and chemical engineering to describe the non-reactive
mass transport in matrix-fracture media [8], in two-region large-scale systems [83]
as well as for the heat transfer [91] in two-phase/region porous media. However, the
complexity of the problem is quite intimidating. Hence, the dilemma of reconciling
Problem 1 and Problem 2 in a non-equilibrium situation appears fundamental.
1.2.3 A one-equation, non-equilibrium model
There has been some interesting work suggesting that it is possible to develop a
one-equation model that applies to non-equilibrium conditions under some timeconstraints. Cunningham and Mendoza-Sanchez in [65] compared the behaviors of
the one-equation model Eq (1.1) (“the simple model”) and the “idealized biofilm”
model. They show that these are equivalent under steady state conditions and “effectively indistinguishable when the rate-controlling process is either external mass
transfer or internal mass transfer” under transient conditions. From a more fundamental perspective, Zanotti and Carbonell showed in [295] that, for the non-reactive
case, two-equation models have a time-asymptotic behavior which can be described
in terms of a one-equation model. The demonstration is based on the moments
matching principle at long times and does not assume local mass equilibrium. They
considered the time-infinite behavior of the first two centered moments of a two-
135
136
INTRODUCTION
equation model developed using the volume averaging with closure theory. The
essential idea here is that the time-asymptotic behavior of a multidomain formulation can be undertaken using a one-equation model even in a non-equilibrium (i.e.,
where the concentrations in the two regions are not at equilibrium relative to one
another) situation.
One could follow the approach of Zanotti and Carbonell to develop an upscaled
theory for the reactive case, but their approach is not straightforward. For example,
in the reactive case it is not possible to adopt a time-infinite limit of the zeroth
order moment. This is primarily because the chemical species is consumed by the
microorganisms and, consequently, its mass tends toward zero. One resolution is
to determine only the long-time rate of consumption, by considering eigenvalue
problems or Laplace transformations. However, this approach leads to a very complex
two-step analysis.
The development of the one-equation time-asymptotic model in one-step would
be a useful development. Dykaar and Kitanidis developed such a technique in [92]
starting directly from the microscale boundary value problem and using a Taylor-ArisBrenner moment analysis; in their analysis they computed the dispersion tensor, the
effective reaction rate and the effective solute velocity. However, there are two areas
in this previous work that could be improved; these are as follows
1. They considered a macroscopic average of the solute concentration only on the
fluid-phase and, while the model does not assume local mass equilibrium, it is
ambiguous as to what the specific model limitations are.
2. The moments matching technique in their analysis [92] makes the assumption
that the behavior of the third and higher spatial moments can be neglected
and that only the smallest eigenvalue of the spatial operator can be considered
to describe the reaction rate. These hypothesis have different meanings in the
single phase configuration and in the multiphase situation. In the work by
Dykaar and Kitanidis, it is unclear how the phase configuration applies to the
analysis.
In the non-reactive case, it has been proven [78] that the the two-step method proposed by Zanotti and Carbonell is strictly equivalent to a one-step technique based
on a particular volume averaging theory presented in that work. The essential feature of that theory, presented in the previous part for a non-reactive situation, is the
definition of a useful but unusual perturbation decomposition. This decomposition
is usually undertaken using fluctuations, conventionally defined in applications to
subsurface hydrology by Gray [118]. For that kind of description, the pointwise concentration is expressed as an intrinsic averaged on the phase plus a perturbation. In a
n − phase system, this decomposition leads to an n-equation macroscopic system
and in our case would lead to the two-equation model previously discussed. Rather
than using an intrinsic averaged on each phase, the perturbation concept can be
1.2 C O N T E X T
extended to a weighted volume averaging hciγω of the pointwise concentration on all
the different phases Eq (1.3), leading to a one-equation model. It is defined as
hciγω =
εγ
εω
hcγ iγ +
hcω iω
εγ + εω
εγ + εω
(1.3)
where εi is the volumic fraction occupied by the i − phase.
In this study, we use this variant of the technique of volume averaging with closure in the reactive case to develop a one-equation model. This model is different
from those based on the local mass equilibrium assumption in that it does not impose specific conditions regarding the concentration in the two phases. Rather, it
requires only that at long times the resulting balance equation is fully described
by an advection-dispersion-reaction type equation, that is, by its first two spatial
moments. This assumption means that the transport process is dispersive, and that
the reactions do not themselves lead to spatial asymmetries for an initially symmetric
solute distribution. This is very much in the spirit of the work by Dykaar and Kitanidis
except that our model Eq (1.4) describes the total mass present in the system and,
hence, exhibits different effective velocity v∗ , dispersion tensor D∗ and reaction rate
α∗ . The constraints associated with the theoretical development are also extensively
discussed.
∂hciγω
+ v∗ · ∇hciγω = ∇ · (
∂t
D∗ · ∇hciγω) − α∗hciγω
(1.4)
The remainder of the part is organized as follows. First, we derive the one-equation
non-equilibrium reactive model. The microscopic equations describing the system
at the pore scale are written and we use the volume averaging upscaling process;
we define a unique fluctuation that is subsequently used to obtain a macroscopic
(but unclosed) one-equation model. Then, we establish a link between the two scales
through a closure problems. Finally, we show that a closed form of the macroscopic
equation can be obtained where effective parameters depend explicitly upon closure
variables solved over a representative cell. We explore numerically some solutions
to the closure problem, and compare the non-equilibrium model to (1) the local
equilibrium model and (2) pore-scale simulations.
137
2
UPSCALING
2.1
M I C R O S C O P I C E Q U AT I O N S
Our study starts with the pore-scale description of the transport of a contaminant/nutrient in the porous medium. In the fluid (γ) phase, convective and diffusive
transport are considered, and it is assumed that there is no reaction. In the biofilm
(ω) phase, only diffusive transport and a reaction are considered. For the purposes of
this paper, the velocity field is assumed to be known pointwise as a vector field. Mass
balanced equations for the biofilm-fluid-solid system takes the following form
∂cγ
+ ∇ · (cγ vγ ) = ∇ · (
∂t
γ − phase :
BC1 :
BC2 :
BC3 :
BC4 :
− (nγσ ·
Dγ · ∇cγ)
Dγ) · ∇cγ = 0
cω = cγ
Dγ) · ∇cγ = − (nγω · Dω) · ∇cω
− (nωσ · Dω ) · ∇cω = 0
− (nγω ·
ω − phase :
∂cω
= ∇·(
∂t
(2.1)
on S γσ
(2.2a)
on S γω
(2.2b)
on S γω
(2.2c)
on S ωσ
(2.2d)
Dω · ∇cω) + R ω
(2.3)
Here, cγ is the chemical species concentration in the γ − phase, and cω is the
concentration in the ω − phase (which can be interpreted as the volume average
concentration in the extracellular space [284, 287]). The symbols γ and ω represent the diffusion tensors in the γ and ω-phases, respectively; R ω is the reaction rate
in the ω-phase, the formulation of this term is detailed in section 2.5; nγω is the unit
normal pointing from the γ-phase to the ω-phase; nγσ is the unit normal pointing
from the γ-phase to the σ-phase; nωσ is the unit normal pointing from the ω-phase
to the σ-phase; S γω is the Euclidean space representing the interface between the
γ-phase and the ω-phase; S γσ is the interface between the γ-phase and the σ-phase;
and S ωσ is the interface between the ω-phase and the ω-phase.
D
D
139
140
UPSCALING
Figure 38: Pore-scale description of a Darcy-scale averaging volume
2.2
AV E R A G E S D E FI N I T I O N S
To obtain a macroscopic equation for the mass transport at the Darcy-scale, we
average each microscopic equation at the pore-scale over a representative region
(REV), V Figs (37) and (38). Vγ and Vω are the Euclidean spaces representing the γand ω-phases in the REV. Vγ and Vω are the Lebesgue measures of Vγ and Vω , that is,
2.2 AV E R A G E S D E FI N I T I O N S
the volumes of the respective phases. The Darcy-scale superficial average of ci (where
i represents γ or ω) is defined the following way
1
hcγ i =
V
Z
1
cγ dV , hcω i =
V
Vγ (x,t)
Z
Vω (x,t)
cω dV
(2.4)
Then, we define intrinsic averaged quantities
1
hcγ i =
Vγ (x, t)
γ
Z
1
cγ dV , hcω i =
Vω (x, t)
Vγ (x,t)
Z
ω
Vω (x,t)
cω dV
(2.5)
Volumes Vγ (x, t) and Vω (x, t) are related to the volume V by
εγ (x, t) =
Vγ (x, t)
Vω (x, t)
, εω (x, t) =
V
V
(2.6)
Hence, we have
hcγ i = εγ (x, t)hcγ iγ , hcω i = εω (x, t)hcω iω
(2.7)
Due to the growth process, the geometry associated with the biofilm-phase can
evolve in time. However, we will assume that changes in the Vγ and Vω volumes
are decoupled from the transport problem. There is substantial support for this
approximation because the characteristic time for growth is much larger than the
characteristic time for transport processes [285, 198]. Moreover, porosities εγ and
εω are also supposed constant in space so that we consider a homogeneous porous
medium.
As stated above, the goal of this part is to devise a one-equation model that describes the evolution of the solute mass in the two phases by a single equation at the
Darcy-scale. Toward that end, we define two additional macroscopic concentrations.
The first is the spatial average concentration, defined by
hci = εγ hcγ iγ + εω hcω iω
(2.8)
The second is a volume-fraction weighted averaged concentration [285, 188]
hciγω =
εγ
εω
hcγ iγ +
hcω iω
εγ + εω
εγ + εω
(2.9)
141
142
UPSCALING
During the averaging process, there arise terms involving the point values for cγ , cω ,
and vγ . To treat these term conventionally, one defines perturbation decompositions
as follows
cγ = hcγ iγ + c̃γ
(2.10)
cω = hcω iω + c̃ω
(2.11)
vγ = hvγ iγ + ṽγ
(2.12)
With these conventional decompositions, the averaging process would lead to the
formulation of a two-equation model, where a separate upscaled equation would be
developed for each phase.
We will adopt a fundamentally different concentration decomposition which allows
the development of a one-equation model that is different from the one-equation
model that assumes local mass equilibrium. To do so, we define the weighted averaged
concentration, hciγω by the decompositions
cγ =hciγω + ĉγ
(2.13a)
cω =hciγω + ĉω
(2.13b)
Notice that with this definition, we do not generally have the condition that the
intrinsic average of the deviation is zero, i.e., hc̃γ iγ , hc̃ω iω = 0. However, we do have
a generalization of this idea in the form
εω hĉω iω + εγ hĉγ iγ = 0
2.3
(2.14)
AV E R A G I N G E Q U AT I O N S
To start, the averaging operators and decompositions defined above are applied to
Eqs (2.1) and (2.3); the details of this process are provided in Appendix A. The result is
γ − phase
∂εγ hcγ iγ
∂t
Z
Z
1
1
γ
+∇ · (εγ hcγ i hvγ i ) = ∇ · εγ Dγ · ∇hcγ i +
nγω c̃γ dS +
nγσ c̃γ dS
Vγ S γω
Vγ S γσ
Z
Z
1
1
+
(nγω · Dγ ) · ∇cγ dS +
(nγσ · Dγ ) · ∇cγ dS − ∇ · hc̃γ ṽγ i
(2.15)
V S γω
V S γσ
γ
γ
2.4 T H E M A C R O S C O P I C C O N C E N T R AT I O N I N A M U LT I P H A S E S Y S T E M
ω − phase
∂εω hcω iω
∂t
Z
Z
1
1
ω
= ∇ · εω ω · ∇hcω i +
nωγ c̃ω dS +
nωσ c̃ω dS
Vω Sωγ
Vω Sωσ
Z
Z
1
1
(nωγ · ω ) · ∇cω dS +
(nωσ · ω ) · ∇cω dS
+
V Sωγ
V Sωσ
D
D
D
+εω hR ω iω
2.4
(2.16)
T H E M A C R O S C O P I C C O N C E N T R AT I O N I N A M U LT I P H A S E S Y S T E M
At this point, one clearly needs two macroscale equations to describe the system
(one-equation for each phase) and it is unclear how the mass transport should be
represented using a single macroscopic concentration. To understand this point, it is
necessary to embrace a more general view of the problem. Experimentally, biofilms
are often studied in laboratory devices such as columns. One may then ask the
questions (1) “What concentration are we measuring at the output of a column
colonized by biofilm ?" and (2) “How do we establish a relationship between this
experimental measure and the concentration in our model ?".
Let us start with the question (1). The concentration measured at the output depends on the design of the column, the physical and chemical properties of the porous
medium and of the flow as well as on the experimental device used for the measure.
For example, let us assume that we are trying to obtain the elution curve of a tracer
(concentration ci in the i−phase) at the output of the column by sampling the water
on relatively small time intervals ∆T (say, a hundred samples for one elution curve)
and then measuring the concentration within each volume. For high microscopic
Péclet numbers, one may measure a quantity close to hcγ iγ as the transport is driven
by the convection in the water-phase. For microscopic Péclet numbers lower than
unity, that is, a transport driven by diffusion, one may measure something closer to
hciγω . In this context, a general definition of the concentration would be
ZtZ
hCi =
G(x − y, t − τ)c(y, τ) dV(y) dτ
(2.17)
0 V
where G is a spatio-temporal kernel corresponding to a weighting function accounting for the measurement device, the column device and the physics of the transport,
and c is the concentration distribution defined by

cγ
in the γ − phase



c =  cω in the ω − phase 
0
in the σ − phase
(2.18)
143
144
UPSCALING
Then, it is necessary to address question (2), that is, we must find a relationship
between hCi and the concentrations appearing in our models. There are two different
ways to proceed. First, it is possible to formulate a generalized volume averaging theory to directly describe the transport of hCi. This has been proposed in [208]. Second,
it is feasible to describe the transport of a relatively simple averaged concentration
and to apply the correct kernel a posteriori. In this case, the concentration to be used
in the model can be chosen on the basis of its relevance from a theoretical point of
view.
In this part, we use hciγω as a macroscopic concentration, that is, we are interested
in following the spatio-temporal macroscopic evolution of the total component mass
in the porous medium. One significant advantage of this definition is that, in a nonreactive medium, the concentration hciγω is conservative; unlike the individual phase
averages, which are not conservative due to interphase mass transfer. For example,
hcγ iγ is often used as a macroscopic concentration but looses many features of the
transport processes. Then, to go back to hCi, one needs to determine the kernel F
defined by
ZtZ
hCi =
F(x − y, t − τ)hciγω (y, τ) dV(y) dτ
(2.19)
0 V
The precise determination of the kernel G or F represents an extremely difficult
task so that, in real problems, one can usually only approximately apprehend the
correct concentration to use for a specific problem. Such descriptions are inevitable
but one must keep in mind that these are approximations.
Going back to the macroscopic equations (2.15) and (2.16), it is then necessary
to make all intrinsic concentrations hci ii disappear. This kind of description can be
developed using the nonconventional decompositions defined by hciγω [205] which
naturally arises when summing Eqs (2.15) and (2.16). We have applied this kind of
analysis to the two macroscale equations developed above; the detailed derivation
can be found in Appendix B. The following non-closed equation results from this
analysis
εγ
εγ
εω
∂hciγω
γω
}
+∇·
hciγω hvγ iγ = ∇ · {
D
+
D
ω
γ · ∇hci
∂t
εγ + εω
εγ + εω
εγ + εω
εγ
εω
ω
γ
+ ∇·
D
·
h∇
ĉ
i
+
D
·
h∇
ĉ
i
ω
ω
γ
γ
εγ + εω
εγ + εω
εω
1
+
hR ω iω −
∇ · hĉγ vγ i
(2.20)
εγ + εω
εγ + εω
2.5 R E A C T I O N T E R M
2.5
REACTION TERM
The form of the reaction rate has not yet been detailed but, at this point, it is important
to make further progress. The classical dual-Monod [170] reaction rate for electron
donor A and acceptor B, is widely adopted to describe biofilm substrate uptake and
growth in systems with a single substrate and a single terminal electron acceptor. In
this case, the reaction rate is given by a hyperbolic kinetic expression of the form
R ω = −α
cBω
cAω
cAω + K A cBω + K B
(2.21)
Here, the α is the substrate uptake rate parameter (often expanded as α = khρb i,
where k is the specific substrate uptake rate parameter, and hρb i is the microbial
concentration; cf. [286]). One often consider the case where the electron acceptor is
not limiting cBω K B , in which case the kinetics take the classical Monod form
R ω = −α
cAω
cAω + K A
(2.22)
which can be written
R ω = −α
cω
cω + K
(2.23)
This is beyond the scope of this paper to propose a technique to upscale such nonlinear kinetics and we will only consider the linear case.
Rω = −
α
K
cω
(2.24)
These linear kinetics can be seen as a particular case of the classical Monod for
which cω K , that is, a highly reactive biofilm or relatively low concentrations. This
approximation has been undertaken incalculable times [244, 41, 36] and is discussed
in [65, 92].
2.6
N O N - C L O S E D M A C R O S C O P I C F O R M U L AT I O N
Introducing linear kinetics in Eq (2.20) leads to
εγ
εγ
εω
∂hciγω
γω
}
+∇·
hciγω hvγ iγ = ∇ · {
D
+
D
ω
γ · ∇hci
∂t
εγ + εω
εγ + εω
εγ + εω
εγ
εω
ω
γ
+ ∇·
D
·
h∇
ĉ
i
+
D
·
h∇
ĉ
i
ω
ω
γ
γ
εγ + εω
εγ + εω
1
1
−
∇ · hĉγ ṽγ i −
∇ · (hĉγ ihvγ iγ )
εγ + εω
εγ + εω
εω
α
(hciγω + hĉω iω )
−
(2.25)
εγ + εω K
145
146
UPSCALING
Although Eq (2.25) represents a macroscale mass transport equation, it is not yet
under a conventional form because deviation concentrations still remain. Eliminating
these deviation concentrations, and hence uncoupling the physics at the microscale
from the physics at the macroscale, is referred to as the closure problem.
3
CLOSURE
3.1
D E V I AT I O N E Q U AT I O N S
To close Eq (2.25), we first need to develop balance equations for the concentration
deviations, ĉγ and ĉω . Going back to their definitions Eqs (2.13) suggests that these
equations can be obtained by subtracting the averaged equation Eq (2.20) to the microscopic mass balanced equations Eqs (2.1) and (2.3). To make further progress, it is
necessary to make some simplifications. We will assume that all the terms containing
only second order derivatives of surface integrated or volume averaged quantities
are negligible compared to spatial derivatives of fluctuation quantities over the REV.
As an aside, terms containing derivatives of averaged quantities are often referred
to as non-local terms. This means that these can not be calculated locally on a REV;
rather, they act as source terms and, if they can not be neglected, that is, if the hypothesis of separation of length scales is not valid, they impose a coupling between
the microscale and the macroscale problems.
Eq (2.1) minus Eq (2.20)
∂ĉγ
+ ∇ · (vγ ĉγ ) =∇ · (
∂t
Dγ · ∇ĉγ) − ∇ · (ṽγhciγω) − ε
εω
∇ · (hvγ iγ hciγω )
+
ε
ω
γ
ε
εω
γ
ω
γ
−∇·
ω · h∇ĉω i +
γ · h∇ĉγ i
εγ + εω
εγ + εω
α εω
1
(hciγω + hĉω iω ) +
+
∇ · hĉγ vγ i (3.1)
K εω + εγ
εγ + εω
D
D
Eq (2.3) minus Eq (2.20)
Dω · ∇ĉω) + ε
εγ
+
ε
γ
ω
εγ
εω
ω
γ
−∇·
ω · h∇ĉω i +
γ · h∇ĉγ i
εγ + εω
εγ + εω
εγ
α
α εω
α
1
−
hciγω +
hĉω iω − ĉω +
∇ · hĉγ vγ i
K εω + εγ
K εω + εγ
K
εγ + εω
∂ĉω
=∇ · (
∂t
D
D
(3.2)
147
148
CLOSURE
We will impose the condition that we are interested in primarily the asymptotic
behavior of the system; thus, we can adopt a quasi-steady hypothesis. In essence,
this constraint indicates that there is a separation of time scales for the relaxation of
ĉγ and ĉω as compared to the time scale for changes in the average concentration,
hciγω . Such constraints can be put in the form
Tγ∗ ∗
Tω
l2γ
lγ
|| γ || hvγ iγ
D
,
l2ω
Dω||
||
,
K
(3.3)
α
∗ ) is a characteristic time associated to ∂ĉγ (respectively
where Tγ∗ (respectively Tω
∂t
∂ĉω
);
||
.
||
is
the
tensorial
norm
given
by
∂t
T
|| || = 12
√
T : T = 21
p
Tij Tji
(3.4)
The vector norm is given by
1
hvγ iγ = (hvγ iγ · hvγ iγ ) 2
(3.5)
This hypothesis is the key to understanding the time-asymptotic behavior of the
model developed herein. It has been shown in [78, 205], in the non-reactive case, that
this quasi-stationarity assumption is equivalent to time-asymptotic models derived
through moments analysis [295] from two-equation models. In other words, the
assumption of quasi-stationarity on the ĉ perturbations is much more restrictive than
the one on c̃ which leads to the two-equation model. One other way of seeing it is to
express ĉ as
ĉi = hĉi ii + c̃i
(3.6)
so that
∂ĉi ∂hĉi ii ∂c̃i
=
+
∂t
∂t
∂t
(3.7)
∂ĉ
∂c̃
∂hĉ ii
Hence, imposing constraints on ∂ti results in constraints on ∂ti but also on ∂ti ;
∂c̃
in opposition to constraints only on ∂ti in the two-equation models. With these
approximations, the closure problems can be rewritten as follows
3.2 R E P R E S E N TAT I O N O F T H E C L O S U R E S O L U T I O N
εω
∇ · (vγ ĉγ ) = ∇ · (Dγ · ∇ĉγ ) − ∇ · (ṽ‚ hciγω ) −
εω + εγ
εγ
εω
ω
γ
− ∇·
Dω · h∇ĉωi + ε + ε Dγ · h∇ĉγi
εγ + εω
γ
ω
εγ
γ
γ
γ
(∇ · hĉγ ṽγ i + hvγ i ∇hĉγ i )
+
εγ + εω
α
εω
(hciγω + hĉω iω )
+
K εω + εγ
(3.8)
− (nγσ · Dγ ) · ∇ĉγ = (nγσ · Dγ ) · ∇hciγω on S γσ
ĉω = ĉγ on S γω
BC3 : − (nγω · Dγ ) · ∇ĉγ = − (nγω · Dω ) · ∇ĉω − {nγω · (Dω − Dγ )} · ∇hciγω on S γω
BC4 : − (nωσ · Dω ) · ∇ĉω = (nωσ · Dω ) · ∇hciγω on S ωσ
BC1 :
BC2 :
εγ
0 = ∇ · (Dω · ∇ĉω ) +
εγ + εω
εγ
εω
ω
γ
− ∇·
Dω · h∇ĉωi + ε + ε Dγ · h∇ĉγi
εγ + εω
γ
ω
εγ
γ
γ
γ
(∇ · hĉγ ṽ‚ i + hvγ i ∇hĉγ i )
+
εγ + εω
εγ
α
εω
α
α
−
hciγω +
hĉω iω −
ĉω
K εω + εγ
K εω + εγ
K
3.2
(3.9a)
(3.9b)
(3.9c)
(3.9d)
(3.10)
R E P R E S E N TAT I O N O F T H E C L O S U R E S O L U T I O N
The mathematical structure of this problem indicates that there are a number of
nonhomogeneous quantities involving hciγω that act as forcing terms. Under the
conditions that a local macroscopic equation is desired, it can be shown (c.f., [282])
that the general solution to this problem takes the form
ĉγ = bγ · ∇hciγω − sγ hciγω
γω
ĉω = bω · ∇hci
− sω hci
γω
(3.11)
(3.12)
Here, the variables bγ , bω , sγ , and sω can be interpreted as integrals of the associated
Greens functions for the closure problem. This closure fails to capture any characteristic time associated to the exchange between both phases in opposition to the closure
used for two-equation models which describes one characteristic time associated
with the exchange. Only non-local theories or direct pore-scale simulations would be
able to recover all the characteristic times involved in this process.
149
150
CLOSURE
Upon substituting this general form into the closure problem, we can collect terms
involving ∇hciγω and hciγω . The result is the following set of coupled closure problems in which derivatives of averaged quantities are neglected
Problem I (s-problem)
∇ · (vγ sγ ) = ∇ · (
Dγ · ∇sγ) − Kα ε
εγ
α
εω
−
hsγ iγ
+
ε
K
ε
+
ε
ω
γ
ω
γ
D
− (nγσ · γ ) · ∇sγ
sω
− (nγω · γ ) · ∇sγ
− (nωσ · ω ) · ∇sω
BC1 :
BC2 :
D
D
BC3 :
BC4 :
0 = ∇·(
=
=
=
=
0
sγ
− (nγω ·
0
Dω) · ∇sω
(3.13)
on S γσ
(3.14a)
on S γω
(3.14b)
on S γω
(3.14c)
on S ωσ
(3.14d)
Dω · ∇sω) + Kα ε
εγ
α εω
α
+
hsω iω − sω
K εω + εγ
K
ω + εγ
(3.15)
Problem II (b-problem)
D
εγ
εω
α
hvγ iγ −
hbγ iγ
εω + εγ
K εω + εγ
εω
+
Dω · h∇sω iω + ε ε+γε Dγ · h∇sγ iγ
εγ + εω
γ
ω
εγ
hsγ vγ iγ
− 2Dγ · ∇sγ + vγ sγ −
εω + εγ
vγ · ∇bγ =∇ · ( γ · ∇bγ ) − ṽγ −
BC1 :
− (nγσ · Dγ ) · ∇bγ
=
(nγσ · Dγ ) (1 − sγ )
BC2 :
bω
=
bγ
=
− nγω · (Dω − Dγ ) (1 − sγ )
BC3 :
BC4 :
− nγω · (Dγ · ∇bγ − Dω · ∇bω )
− (nωσ · Dω ) · ∇bω
=
(nωσ · Dω ) (1 − sω )
(3.16)
on S γσ
(3.17a)
on S γω
(3.17b)
on S γω
(3.17c)
on S ωσ
(3.17d)
0 =∇ · (Dω · ∇bω ) +
εγ
α
εω
α
hvγ iγ +
hbω iω −
bω
εγ + εω
K εω + εγ
K
εω
+
Dω · h∇sω iω + ε ε+γε Dγ · h∇sγ iγ − 2Dω · ∇sω − ε ε+γ ε hsγ vγ iγ
εγ + εω
γ
ω
ω
γ
(3.18)
There is an interesting discussion concerning the concept of representative elementary volume (REV) which is often misunderstood. Within hierarchical porous media,
there is substantial redundancy in the spatial structure of the transport processes at
the microscale, that is, the information needed to calculate the effective parameters is
contained in a relatively small representative portion of the medium. Within this REV,
internal boundary conditions Eqs (3.17) between the different phases are determined
by the physics at the pore-scale. However, in order to ensure unicity of the s and
b fields, it is also mandatory to adopt a representation for the external boundary
condition between the REV and the rest of the porous medium. This condition is not
3.3 C L O S E D M A C R O S C O P I C E Q U AT I O N
determined by the physics at the pore-scale but rather represents a way of closing
the problem. At first, it is unclear how this choice should be made and it results in
a significant amount of confusion in the literature. From a theoretical point of view,
if the REV is large enough (read if the hypothesis of separation of length scales is
verified), it has been shown [274] that effective parameters do not depend on this
boundary condition.
In the real world, this constraint is never exactly satisfied, that is, the boundary
condition can substantially influence the microscopic fields. However, it is important
to notice that in the macroscopic Eq (2.25), ĉ appear only under integrated quantities.
Because of this, the dependence of effective parameters upon the solution of the
closure problem is essentially mathematically of a weak form [190]. Hence, one
could choose, say, Dirichlet, Neumann, mixed or periodic boundary conditions to
obtain a local solution which produces acceptable values for the associated averaged
quantities. As previously discussed in the literature [268, 93, 216, 199, 51], the periodic
boundary condition lends itself very well for this application as it induces very little
perturbation in the local fields, in opposition to, say, Dirichlet boundary conditions.
It must be understood that this does not mean that the medium is interpreted as
being physically periodic. For the remainder of this work, we will assume that the
medium can be represented locally by a periodic cell Eq (3.19) and that the effective
parameters can be calculated over this representative part of the medium.
Periodicity :
ĉi (x + lk ) = ĉi (x)
k = x, y, z
(3.19)
Periodicity :
bi (x + lk ) = bi (x)
k = x, y, z
(3.20)
Periodicity :
si (x + lk ) = si (x)
k = x, y, z
(3.21)
We also have Eqs (3.20) and (3.21)
In these equations Eqs (3.19), (3.20) and (3.21), we have used lk to represent the three
lattice vectors that are needed to describe the 3-D spatial periodicity.
In addition to these periodic boundary conditions, one usually needs to impose
constraints on the intrinsic averaged of the closure fields in order to ensure unicity
of the solutions. To find out these additional equations, we use hc̃γ iγ , hc̃ω iω = 0.
In our case, this is not necessary to constrain the fields because the reactive part
of the spatial operator ensures, mathematically, unicity of the solutions. However,
numerical computations, in situations where the reaction has little importance in
comparison to other processes, can lead to some discrepancies. To avoid this problem,
it is important to impose εω hĉω iω +εγ hĉγ iγ = 0, that is, εω hbω iω +εγ hbγ iγ = 0 and
εω hsω iω + εγ hsγ iγ = 0.
3.3 C L O S E D M A C R O S C O P I C E Q U AT I O N
Substituting Eqs (3.11) and (3.12) into Eq (2.25) leads to
151
152
CLOSURE
∂hciγω
+ v∗ · ∇hciγω = ∇ · (
∂t
D∗ · ∇hciγω) − α∗hciγω
(3.22)
where the effective parameters are given by
v∗
=
D∗
=
α∗
=
D
D
εγ
εω
α
ω
(hvγ iγ + γ · h∇sγ iγ − hsγ vγ iγ ) +
hbω iω
(3.23)
ω · h∇sω i +
εγ + εω
εγ + εω
K
εγ
εω
[ γ · (I − Ihsγ iγ + h∇bγ iγ ) − hvγ bγ iγ ] +
[ ω · (I − Ihsω iω + h∇bω iω )] (3.24)
εγ + εω
εγ + εω
εω
α
(1 − hsω iω )
(3.25)
K ε γ + εω
D
D
For simplicity, the porosities are taken to be constants. If the model is applied
to media with non constant porosities, one should take care to consider gradients
of ε (c.f., Appendixes). Moreover, the macroscopic equation is written under a nonconservative form so that it exhibits only effective velocity, dispersion and effective
reaction rate. It is convenient to write it this way for the purpose of comparing the
asymptotic model with other models.
However, notice that a more general conservative expression would be
∂hciγω
+∇·
∂t
D
εγ
hciγω hvγ iγ = ∇ · ( ∗c · ∇hciγω ) − ∇ · (d∗c hciγω ) − v∗c · ∇hciγω − α∗c hciγω (3.26)
εγ + ε ω
D
d∗
c=
v∗
c=
D
∗=
c
α∗c =
D
D
εγ
εω
γ
γ
ω
εγ +εω ( γ · h∇sγ i − hsγ vγ i ) + εγ +εω ( ω · h∇sω i )
εω
α
ω
εγ +εω K hbω i
(3.27)
(3.28)
D
εγ
εω
γ
γ
γ
ω
ω
εγ +εω [ γ · (I − Ihsγ i + h∇bγ i ) − hvγ bγ i ] + εγ +εω [ ω · (I − Ihsω i + h∇bω i )] (3.29)
εω
α
ω
(3.30)
K εγ +εω (1 − hsω i )
Notice that, at this point, if v∗c plays mathematically the role of a velocity, it is
directly linked to the chemical reaction and should not be discarded if one considers
non-convective flows.
4
N U M E R I C A L R E S U LT S
Figure 39: Total geometry
Ideally, one could compare the theory developed above with the results of direct
experimental measurements conducted at both the microscale and at the macroscale.
Theoretically, it is possible to obtain a three dimensional image of the three phases
biofilm-liquid-solid. This is an area of active research [229], and workers are continuing to develop methods such that the microscale structure of a biofilm within a
porous medium can be measured [130, 76]. Currently, however, the results from such
multi-scale experimental measurements are not available.
The goal of this section, then, is to provide some characteristic features of the model
previously devised on a simplified 2D medium Figs (39) and (40) using numerical
methods. We adopt a conceptual construction which captures the main physics of
the problem. In Fig (40), σ − phase is represented by solid black, the γ − phase is
given by light grey, and the grey lies for the ω − phase. One should notice that at the
macroscale only a 1D model is needed for this particular geometry. For both the 2D
and 1D models, the output boundary condition is set to free advective flux. For the
purposes of this study, 1) we obtain the velocity field by solving Stokes equations,
with no-slip conditions on lateral boundaries, over the entire system, 2) we will only
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154
consider a spheric diffusion tensor for the biofilm and water-phases 3) we fix K = 0.5
ω
and DΣ = D
Dγ = 0.3 and take lγ = 0.5.
Figure 40: Representative cell
For these simulations, we have the following goals
1. To establish the behavior of the effective parameters as functions of Péclet and
Damköhler numbers.
2. To compare these effective parameters with those of the local mass equilibrium
model, as developed in [114].
3. To validate the model against pore-scale simulations both stationary and transient.
All the numerical calculations were performed using the COMSOLT M Multiphysics
package 3.5 based on a finite element formulation. For the resolution of Stokes equation, we use quadratic Lagrange elements for the velocities and linear elements for the
pressure. For the resolution of the advection-diffusion equations, we use a quadratic
Lagrange element formulation. Residuals are computed using a quadrature formula
of order 2 for linear Lagrange elements and 4 for quadratic Lagrange elements. The
linear systems are solved using the direct solver UMFPACK based on the Unsymmetric
MultiFrontal method.
The problem Eqs (2.1)-(2.3) at the pore-scale can be rewritten under the following
dimensionless form
∂cγ0
0 0
+
P
e
∇
·
c
v
= ∆cγ0
γ
γ
∂t 0
γ − phase :
−nγσ · ∇cγ0 = 0
BC1 :
0
c ω = cγ0
−nγω · ∇cγ0 = − Σ nγω · ∇c 0 ω
0
Σ nωσ · ∇c ω = 0
BC2 :
BC3 :
−D
BC4 :
ω − phase :
D
(4.1)
on S γσ
(4.2a)
on S γω
(4.2b)
on S γω
(4.2c)
on S ωσ
(4.2d)
0
∂cω
0
= ∇ · (DΣ ∇c 0 ω ) − D aDΣ cω
∂t 0
(4.3)
where the normalized concentrations and velocity are given by
c 0ω
= ccω0
(4.4)
c 0γ
cγ
c0
(4.5)
=
v
v 0 γ = hv γiγ
γ
(4.6)
The concentration c0 is the input concentration. Notice that we impose c0 K
which is a sufficient constraint for the linearization of the reaction rate. We have
adopted the following additional definitions for dimensionless quantities
t0 =
thvγ iγ
lγ
(4.7)
The ratio between the diffusion coefficients in the ω − phase and γ − phase is
DΣ
= DDω
γ
(4.8)
The Péclet and Damköhler numbers are specified by
Da
αl2
= K Dγω
Pe =
(4.9)
hvγ iγ lγ
Dγ
(4.10)
The closure problems Eqs (3.13)-(3.18) take the form in dimensionless quantities
Problem I (s-problem)
P e ∇ · (v 0 γ sγ ) = ∆sγ − D a
εγ
εω
− Da
hsγ iγ
εω + εγ
εω + εγ
(4.11)
155
156
−nγσ · ∇sγ = 0
on S γσ
sω = s γ
on S γω
−nγω · ∇sγ = −DΣ nγω · ∇sω
on S γω
BC4 :
−{D}Σ nωσ · ∇sω = 0
on S ωσ
Periodicity :
si (x + lx ) = si (x)
εγ
εω
0 = ∇ · (DΣ ∇sω ) + D a
+ Da
hsω iω − D a sω
εω + εγ
εω + εγ
BC1 :
BC2 :
BC3 :
Problem II (b-problem)
P e v 0 γ · ∇b 0 γ − v 0 γ sγ + ṽγ0 = ∇ · (∇b 0 γ ) − 2∇sγ
(4.12a)
(4.12b)
(4.12c)
(4.12d)
(4.12e)
(4.13)
(4.14)
εγ
Pe
hb 0 γ iγ −
hsγ v 0 γ i
εω + εγ
εω + εγ
εγ
εω
ω
D
h∇sγ iγ
Σ h∇sω i +
εγ + εω
εγ + εω
− Da
+
−nγσ · ∇b 0 γ = nγσ (1 − sγ )
BC2 :
b 0ω = b 0γ
BC3 : −nγω · (∇b 0 γ − DΣ ∇b 0 ω ) = −nγω (DΣ − 1) (1 − sγ )
BC4 :
−nωσ · ∇b 0 ω = nωσ (1 − sω )
Periodicity :
b 0 i (x + lx ) = b 0 i (x)
BC1 :
on S γσ (4.15a)
on S γω (4.15b)
on S γω (4.15c)
on S ωσ (4.15d)
(4.15e)
0 = ∇ · (DΣ ∇b 0 ω ) − 2DΣ ∇sω
εω
Pe
+ Da
hb 0 ω iω − D ab 0 ω −
hsγ v 0 γ i
εω + εγ
εω + εγ
εγ
εω
ω
+
D
h∇sγ iγ
Σ h∇sω i +
εγ + εω
εγ + εω
(4.16)
where
ṽγ0 =
ṽγ
hvγ i
b 0γ =
γ
bγ
lγ
b 0ω =
bω
lγ
(4.17)
Notice that, if one fixes DΣ , the set of equations only depends upon P e and D a.
4.1
D I S P E R S I O N , V E L O C I T Y A N D R E A C T I V E B E H AV I O R
In this section, we solve the closure parameters problems Eqs (4.11)-(4.16) over the
cell Fig (40) for large ranges of P e and D a. Then, the associated effective parameters
are computed and presented Fig (41) for the longitudinal dispersion D∗xx normalized
εγ
ω
Dγ + εγε+ε
Dω; Fig (42) for the x-component of the effective velocity v∗x
with εγ +ε
ω
ω
ε
γ
hvγ iγ ; Fig (43) for the effective kinetics α∗ normalized with
normalized with εγ +ε
γ
εω α
εγ +εγ K .
4.1 D I S P E R S I O N , V E L O C I T Y A N D R E A C T I V E B E H AV I O R
Figure 41: Normalized longitudinal dispersion of the non-equilibrium model as a function of
P e and D a numbers
Figure 42: Normalized x-component of the effective velocity of the non-equilibrium model as
a function of P e and D a numbers
As an aside, notice that for low P e and high D a, the length scales constraints
needed to develop the closed macroscopic equation are not satisfied. However, the
157
158
constraints are developed in terms of order of magnitudes so that it is not clear what is
the exact limit between the homogenizable and non-homogenizable zones. Because
of this, the entire space of P e, D a parameters is kept and we do not clearly establish
this frontier.
Figure 43: Normalized effective reaction rate of the non-equilibrium model as a function of
P e and D a numbers
Effective parameters strongly depend upon the type of boundary conditions at the
small scale, that is, the dispersion in a, say, Dirichlet bounded system or in a Neumann
bounded problem may drastically change. In our case, for low D a, the boundary between the biofilm-phase and the fluid-phase is a flux continuity whereas, as D a tends
toward infinity, the concentration in the biofilm-phase and at the boundary tends toward zero, which can be interpreted, for conceptual purposes, as a Dirichlet boundary
condition. In other words, when P e 1 and D a 1 the medium can conceptually
be represented by a zero-concentration layer surrounding the biofilm, that is, the
substrate reaches a limited interstitial space corresponding to the maxima of the local
velocity field. It results in an increase of the apparent velocity with both P e and D a
numbers. The dispersion exhibits the classical form, and increases mainly with the
P e number as the hydrodynamic dispersion becomes predominant. However, the
log scale hides the fact that D∗ actually depends also on D a and this is presented in
Fig (44). We observe a drastic reduction of the longitudinal dispersion at high D a and
it is in agreement with previous studies [143, 92, 233]. The physics underlying this
effect is reminiscent to the one causing an augmentation of the apparent velocity. As
4.1 D I S P E R S I O N , V E L O C I T Y A N D R E A C T I V E B E H AV I O R
a boundary layer in which cγ = 0 surrounds the biofilm-phase, the solute is confined
in a small portion of the fluid-phase or, more precisely, undertakes biodegradation as
soon as it reaches the edges of this zone. The molecules of solute far away from the
entrance have not visited the entire γ − phase but rather a narrow central portion in
which the fluctuations of the velocity field are limited. As a consequence, the substrate
spreading due to hydrodynamic dispersion (reminiscent to the Taylor dispersion in a
tube) is reduced. Additionally, it is important to keep in mind that our model porous
medium does not exhibit any transverse dispersion. If it was to be considered, one
would expect a different behavior for the transverse dispersion (see discussions in
[143, 92, 233]).
Figure 44: Normalized dispersion behavior of the non-equilibrium model for P e = 1000 as a
function of D a number
The effective reaction rate depends almost only on D a because the mass transfer
through the boundary is mainly driven by diffusion. It seems that, for low D a, the
reaction rate is maximum and decreases when the consumption is too elevated compared to diffusion. It suggests that the reaction rate could be written under the form
ηRmax with η 6 1 a function of D a. Notice that Dykaar and Kitanidis [92], following
the work of Shapiro [233] for a surface reactive medium, also established a theoretical framework for this kind of effectiveness factor using the moments matching
technique. However, their model describes an averaged concentration only on the
water-phase rather than the total mass present in the porous medium at a given time.
The reader is referred to the section 2.4 for an extensive discussion of this point.
159
160
Figure 45: Relative differences of the longitudinal dispersion between local mass equilibrium
and non-equilibrium models as functions of P e for different D a numbers
4.2
R E L AT I O N S H I P W I T H T H E L O C A L M A S S E Q U I L I B R I U M M O D E L
In this part, we compare the local mass equilibrium model as developed in [114] with
the non-equilibrium one-equation model. The local mass equilibrium model takes
the form
∂hciγω
+ v∗ Equ · ∇hciγω = ∇ ·
∂t
D∗Equ · ∇hciγω
− α∗Equ hciγω
(4.18)
v∗Equ =
D∗Equ =
α∗Equ
εγ
εγ +εω
[Dγ h∇bγEqu iγ ] +
=
εγ
γ
εγ +εω hvγ i
εω
1
γ
εγ +εω [ ω h∇bγEqu i ] − εγ +εω hṽγ bγEqu i
εω
α
K εγ +εω
D
(4.19)
(4.20)
(4.21)
The closure parameters are solutions of the following problem
γ − phase :
P e v 0 γ · ∇b 0 γEqu + ṽγ0 = ∇ · {∇b 0 γEqu }
(4.22)
4.3 C O M P A R I S O N W I T H D I R E C T N U M E R I C A L S I M U L AT I O N
−nγσ · ∇b 0 γEqu = nγσ
on S γσ
0
0
BC2 :
b ωEqu = b γEqu
on S γω
0
0
BC3 :
−nγω · (∇b γ − DΣ ∇b ωEqu ) = −nγω (DΣ − 1) on S γω
BC4 :
−nωσ · ∇b 0 ωEqu = nωσ
on S ωσ
0
0
Periodicity :
b iEqu (x + lx ) = b iEqu (x)
BC1 :
ω − phase :
0 = ∇ · {DΣ ∇b 0 ωEqu } − D ab 0 ωEqu
(4.23a)
(4.23b)
(4.23c)
(4.23d)
(4.23e)
(4.24)
where
b 0 γEqu =
bγEqu
lγ
b 0 ωEqu =
bωEqu
lγ
(4.25)
In the local mass equilibrium model, the effective reaction rate and velocity are
constant in terms of P e and D a numbers. Notice that, on Fig (42) and Fig (43) the
effective parameters of the non-equilibrium model are directly normalized with those
of the equilibrium one. Because of this, the comparison is straightforward and both
are close to each other for D a 6 1.
For the dispersion, the relative differences between both models, in terms of P e
and D a numbers, are presented in Fig (45). For P e 6 1, the relative difference is
close to zero so that both models are equivalent for D a 6 1 and P e 6 1. It has been
shown in [114] that this region of the D a, P e space represents the entire region of
validity of the local mass equilibrium model. As a direct consequence, it turns out that
the non-equilibrium model includes the equilibrium one when this one is valid. A
thorough study of the transient behavior for both models is performed and discussed
in the next section.
4.3
C O M P A R I S O N W I T H D I R E C T N U M E R I C A L S I M U L AT I O N
The aim of this section is to provide direct evidences that the model allows a good
approximation of the situation at the pore-scale, to catch its limits and to study some
physics of the problem.
On the one hand, we solve the entire 2D microscopic problem on a total length
of 120Lc (called DNS for Direct Numerical Simulation). On the other hand, we solve
the 1D upscaled models on a total length of 120Lc . First, we observe the stationary
response of the system for different Péclet and Damköhler numbers. Notice that
both boundary conditions at the output are free advective flux. Then, we study the
breakthrough curves at 20Lc , 60Lc and 100Lc for a square input of a width of δt 0 = 5
starting at t 0 = 0 for different Péclet and Damköhler numbers.
161
162
4.3.1 Stationary analysis
Figure 46: DNS and non-equilibrium model stationary concentration fields for P e = 10 and
D a = 10 using normalization on the first cell (DNSB) and on the cell number 20
(DNSA)
The comparison of the concentration fields between the DNS and the non-equilibrium
model is presented Fig (46) for P e = 10, D a = 10; Fig (47) for P e = 100, D a = 100
and Fig( 48) for P e = 1000, D a = 1000. Notice that each circle, cross and square
represents the value of hciγω integrated on a cell.
We solve the stationary boundary value problem Eqs (2.1)-(2.3) with an input
Dirichlet boundary condition of amplitude c0 . Then the results are normalized using
the value of hciγω calculated on the first cell (DNSB) and on the cell number 20
(DNSA) for Fig (46) and Fig (47); and on the first cell (DNSB), on the cell number 20
(DNSA) and 40 (DNSC). The origin of the spatial base is modified consequently to
make them all start at 0.
For all the different situations, the model provides a very good approximation of
the physics at the pore-scale. As previously discussed, the non-equilibrium model
is time-constrained because of the hypothesis of quasi-stationarity on ĉi . Global
stationarity represents a special time-constrained situation for which this hypothesis
is very well satisfied.
Figure 47: DNS and non-equilibrium model stationary concentration fields for P e = 100 and
D a = 100 using normalization on the first cell (DNSB) and on the cell number 20
(DNSA)
However, for P e = 100, D a = 100 and P e = 1000, D a = 1000 some discrepancies
arise between the different normalizations. If the concentration is normalized using
hciγω calculated on a cell far away from the input boundary, the results are closer
between the DNS and the homogenized model. In the DNS, by imposing a Dirichlet
boundary input, we impose c̃γ = 0 and this can not be captured by the macroscopic
model. As a consequence, the flux on S γω is overestimated in the DNS on the first
cells compared to the homogenized model. When P e = 10, D a = 10, this overestimation does not even reach the second cell. For P e = 100, D a = 100, it start to exceed
the first cell. For P e = 1000, D a = 1000, the discrepancy propagates very far from
the boundary input as even the normalization on the cell number 20 does not give
a satisfying result compared to the one on the cell number 40. These discrepancies
appear because of the specific ordering of the porous medium and would not propagate so far from the input in a disordered medium. This question of the impact of
boundary conditions on the comparison between direct numerical simulations and
macroscopic predictions has received some attention in the literature [202, 17, 259].
Corrections of the macroscale boundary conditions, or mixed microscale/macroscale
163
164
approaches are available [17, 259, 40] but this is beyond the scope of this paper to
develop such techniques.
Figure 48: DNS and non-equilibrium model stationary concentration fields for P e = 1000
and D a = 1000 using normalization on the first cell (DNSB), on the cell number 20
(DNSA) and on the cell number 40 (DNSC)
4.3.2 Transient analysis
In this subsection, we study the transient behaviors of the one-equation non-equilibrium
and equilibrium models for a square input of width δt 0 = 5 starting at t 0 = 0. Concentrations are normalized to the amplitude of the square input and the time t 0
is normalized with the characteristic time associated to the advective term. Notice
that, in the transient case, we can not avoid the problem previously presented as the
concentration can not be renormalized straightforwardly.
On Fig (49), the three homogenized models provide a very good approximation of the transport problem. At low Péclet, low
Damköhler numbers, time and space non-locality tend to disappear because time
and length scales are fully separated. Meanwhile, some very little discrepancy, probaI N FL U E N C E O F T H E P É C L E T N U M B E R
Figure 49: Transient breakthrough curves for the DNS, the local non-equilibrium and equilibrium models for a square input of δt 0 = 5 for P e = 1 and D a = 10−5 after a) 20Lc ;
b) 60Lc ; c) 100Lc
bly due to the flux overestimation discussed in the section 4.3.1, exists at the peaks.
The signal even propagates slowly enough for the local mass equilibrium assumption
to be valid.
When the Péclet number reaches values around 100, the local mass equilibrium
assumption becomes clearly inappropriate. Fig (50) shows that the local mass equilibrium model gives a poor approximation of the signal whereas the non-equilibrium
one is still in good agreement. The fact that the peaks for 20Lc are not in such a
good agreement is characteristic of non-locality. Memory functions (convolutions) or
two-equation models should be considered in this case. However, when the signal
spreads, non-locality tends to disappear and the breakthrough curves are in very
good agreement.
For Péclet numbers around 1000, Fig (51), we show that there are some huge
discrepancies between both homogenized model and the DNS, especially at 20Lc
because of the strong non-locality. However, for long times, the non-equilibrium
model seems to recover the tailing and the peak of the signal. Results suggest that
the one-equation local non-equilibrium model might represent, in cases such as
intermediate Péclet numbers or time-asymptotic regime, a good compromise, in
165
166
Figure 50: Transient breakthrough curves for the DNS, the local non-equilibrium and equilibrium models for a square input of δt 0 = 5 for P e = 100 and D a = 10−5 after a)
20Lc ; b) 60Lc ; c) 100Lc
terms of computational demand, between fully transient theories and the local mass
equilibrium model. The importance of non-locality is also emphasized and becomes
particularly obvious in the high Péclet number situation.
When P e = 100 and D a = 100 Fig
(52) and when P e = 1000 and D a = 1000 Fig (53), the local mass equilibrium model
obviously does not recover the total mass of the system, that is, the reaction rate
is overestimated. The non-equilibrium model is much more correct on this aspect.
Meanwhile, at 20Lc it fails to capture non-locality and some discrepancies remain
even when the signal spreads at 60Lc and 100Lc unlike for the low D a situation.
This difference probably comes from the overestimation, in the DNS, of the flux on
the first cells. In the non-reactive case, this effect has very little influence on the
breakthrough curves whereas it is of special importance for high D a situation as the
mass overexchanged disappears. However, notice that, unlike the situation P e =
1000 and D a = 10−5 , the one-equation non-equilibrium model recovers correctly the
shape of the signal. It suggests that in the highly reactive case, the long-time regime
I N FL U E N C E O F T H E D A M K Ö H L E R N U M B E R
4.4 C O N C L U S I O N S C O N C E R N I N G T H E N U M E R I C A L S I M U L AT I O N S
Figure 51: Transient breakthrough curves for the DNS, the local non-equilibrium and equilibrium models for a square input of δt 0 = 5 for P e = 1000 and D a = 10−5 after a)
20Lc ; b) 60Lc ; c) 100Lc
may be adhered quicker than in the low reactive case despite the shift coming from
the input boundary discrepancies.
4.4
C O N C L U S I O N S C O N C E R N I N G T H E N U M E R I C A L S I M U L AT I O N S
First, we study all the effective parameters as functions of P e and D a numbers.
We show that the dispersion exhibits differences between the reactive and the nonreactive case and this is coherent with other studies [143, 92, 233]. Effective reaction
rate and velocities are also presented and we show that they mainly depend upon the
D a number. We also show that the non-equilibrium model includes the local mass
equilibrium one when the conditions of validity of this model are satisfied. From
a theoretical point of view, one should realize that for this special case, the quasistationary analysis is the same for both the Gray decomposition and the non-zero
averaged decomposition since the local mass equilibrium assumption means c̃γ = ĉγ
and c̃ω = ĉω .
167
168
Figure 52: Transient breakthrough curves for the DNS, the local non-equilibrium and equilibrium models for a square input of δt 0 = 5 for P e = 100 and D a = 100 after a)
20Lc ; b) 60Lc ; c) 100Lc
Then, by comparison with direct numerical simulations at the pore-scale, we
show that the model is perfectly adapted to stationary analysis since this represents
a special time-constrained case for which the quasi-stationarity on ĉi is very well
satisfied. We also establish the following limitations
• The model fails to capture very small time phenomena. Two-equations models or fully transient theories may be required in this case. The underlying
consequence is that domains of validity for the different models must need a
time dimension and not only dimensionless parameters such as P e and D a
numbers.
• We emphasize that important discrepancies, coming from the boundary conditions, can propagate through the entire system for high P e and high D a
numbers. Although, it might not propagate so far in disordered media, it requires further investigation.
There are two additional constraints which require supplementary research. The
first one concerns the assumption on the reaction rate. Herein, we suppose that the
4.4 C O N C L U S I O N S C O N C E R N I N G T H E N U M E R I C A L S I M U L AT I O N S
Figure 53: Transient breakthrough curves for the DNS, the local non-equilibrium and equilibrium models for a square input of δt 0 = 5 for P e = 1000 and D a = 1000 after a)
20Lc ; b) 60Lc ; c) 100Lc
concentration of the solute is relatively small, that is, we can consider only linear
kinetics. Upscaling of non-linear Monod type reaction rate in a general framework
is an area of active research. The second one deals with the assumption that the
effective parameters can be calculated on a REV in which the geometry of the biofilm
is fixed. For example, in a real medium, one may have to consider fluctuations of the
porosities or variations of the representative geometry and it is unclear how these
would affect the domain of validity of the time-asymptotic model.
169
5
DISCUSSION AND CONCLUSIONS
5.1
R E L AT I O N T O O T H E R W O R K S
In this work, we derive a one-equation non-equilibrium model for solute transport
in saturated and biologically reactive porous media. Undertaking the description of
multiphase reactive transport using a single one equation approximation has been
done countless times by experimenters. However, very few works have focused on
developing a theoretical basis and on addressing the validity of this approach. Two
situations allowing such a description have been identified in the past and are clarified
in our study. On the one hand, when gradients within the bulk phases are relatively
small, a single partial differential equation on the concentration in the water-phase
can be used to described the mass transport. This situation is often referred to as the
local mass equilibrium condition and has been extensively discussed in [114, 19]. On
the other hand, it has been suggested that a completely different type of constraint
can be formulated to allow a description with merely one equation. Cunningham
and Mendoza-Sanchez in [65] have shown that the one-equation model is strictly
equivalent to the multicontinuum approach under steady state conditions. This
behavior has also been proved many years before in the non-reactive case by Zanotti
and Carbonell in [295].
Our analysis can be seen as an extension and a complement of the works by Cunningham and Mendoza-Sanchez [65] and by Dykaar and Kitanidis [92]. In comparison
with the work by Cunningham and Mendoza-Sanchez, we provide a direct link between the microscopic processes and the macroscale. Our development is based
on the calculation of the effective parameters on a representative volume possibly
accounting for very complex geometries. In addition to the approach by Dykaar and
Kitanidis, we propose to take into account the total mass in the system rather than
just the mass in the water-phase. The strategy adopted in [92] is clearly an extension
of the work by Shapiro and Brenner [233] but additional constraints are necessary
in the multiphase situation and this is not clearly emphasized. The domain of validity of our model is clearly established on the basis of (1) comparisons between
the upscaled results and the direct numerical simulations at the pore-scale and (2)
thorough discussions concerning the first-order closure and the quasi-stationarity of
the problems on the perturbations.
171
172
DISCUSSION AND CONCLUSIONS
Concerning the technique itself, we use a ĉ decomposition for concentrations
which is a more general case of the one introduced in [205] for mass transport and in
[178] for heat transfer. We start our analysis with the microscale description of the
medium and then average the equations to obtain a Darcy-scale description of the
medium. In short times, the multidomain approach provides a better approximation
of the transport processes because, the closure on the c̃ fluctuations captures more
characteristic times and because, as previously discussed, the quasi-stationarity of
the perturbation problem on ĉ is much stronger than the one on c̃. In the non-reactive
situation, these theoretical aspects have been extensively described in [295] and in
[78] but this is the first application to a reactive situation.
5.2
GENERAL CONCLUSIONS
In past research, the calculation of effective parameters has been largely undertaken
using tracer techniques and inverse optimization on the basis of simple heuristic
models. The main problems with this approach are that (1) the macroscale equations
are elaborated on the basis of simple conceptual schemes and it is unclear how much
information these models are able to capture, (2) the effective parameters, say the
dispersion, are often considered as intrinsic to a medium and not recalculated every
time a physical parameter such as the Damköhler number is modified and (3) there is
no clear relationship between the definition of the macroscopic concentrations and
the concentration measured.
Given the advances in terms of imaging techniques and of understanding of the
transport processes, we believe that deterministic upscaling represents an alternative
in many cases. For example, the volume averaging theory lends itself very well for the
exploration of the physics of the transport as well as for the expression of the effective
parameters as a function of the microscale processes on a representative volume.
In conclusion, we provide a solid theoretical background for the one-equation
model along with (1) constraints concerning its validity, (2) a method for the calculation of the effective parameters and (3) a precise definition of the macroscopic
concentration. When applying the model to experimental results, these three points
should be carefully examined.
Part V
CONCLUSIONS, ONGOING WORK AND
PERSPECTIVES
1
CONCLUSIONS
Here, we only briefly summarize the different conclusions specific to each part of
the thesis. We are primarily interested in the global strategy, and focus, in the next
section, on ongoing work and perspectives.
In this thesis, our goal is to understand and model transport phenomena in porous
media with biofilms. The work presented is based on experimental, theoretical and
numerical analyses that provide an interesting novel framework for the study of these
complex systems. First, we present a method for imaging biofilm within opaque
porous media, in part II. This technical breakthrough, entirely developed during the
thesis, uses X-ray computed microtomography, in conjunction with a selection of
specific radiocontrast agents, to obtain a three-dimensional reconstitution of the
biofilm/porous-structure (water-phase, biofilm-phase and grain-phase) with a voxel
size down to 9 µm. This approach provides a promising framework to study the growth
of biofilms within porous structures as well as the response of the microorganisms to
different stimuli and environmental conditions.
On the basis of this information, we propose a modeling strategy for the transport of
solutes in porous media with biofilms. As discussed in the part I, this MVMV approach
follows four steps:
1. image pore-scale biofilms growth within porous structures and formulate equations that describe the various pore-scale phenomena.
2. incorporate these components in efficient pore-scale models based, for example, on Lattice–Boltzmann formulations (e.g., [116]) and validate the proposed
mathematical descriptions.
3. upscale the set of differential equations (from the pore-scale to the Darcy-scale,
eg. parts III and IV) in order to obtain different macroscale models and calculate
effective parameters on the basis of 3-D realistic geometries.
4. develop the domains of validity of the various upscaled models, and validate
the theoretical analysis against Darcy-scale experiments.
In this strategy, the equations that are used at the microscale are validated by comparing direct numerical results (cellular automata or individual-based modeling) against
175
176
CONCLUSIONS
three-dimensional observations. In addition, upscaled models are also studied and
validated against Darcy-scale experiments. While such an approach would be extremely interesting, fully performing the four different steps remains very complex
and this problem has not been tackled yet. In our study, we use a shortcut to this approach, that is, we consider a theoretical situation for which the pore-scale equations
are well-known. On this basis, we devise a macrotransport theory for the transport
of non-reactive as well as biodegraded solutes within porous media colonized by
biofilms. To filter information from the pore-scale/biofilm-scale, we use the classical
volume averaging with closure theory, a new average/fluctuation decomposition and
the spatial moments matching technique. Various Darcy-scale models are developed,
along with their domains of validity and numerical illustrations. If other microscale
linear situations were to be considered, it would be possible to adapt this theory in
order to develop the corresponding macroscale equations.
Ongoing works include the development of numerical tools to calculate the effective parameters on the 3-D image conjointly with other experimental and theoretical
developments. These perspectives are presented in the next sections.
2
ONGOING WORK AND PERSPECTIVES
2.1
NUMERICAL PERSPECTIVES
2.1.1 Calculation of effective properties
While the X-ray tomography technique only captures the spatial distribution of phases
with contrasted attenuation coefficients, numerical calculations can be used in order
to recover more information. For example, one can directly solve Stokes equations on
the images captured by the X-ray tomography technique to obtain the velocity field.
Two examples are given on Fig. (54) . We use the images obtained before and after
biofilm growth on a single polyamide bead. A finite volume formulation is used based
on Uzawa’s algorithm (e.g. [206]). Boundary conditions consist in a pressure gradient
from the bottom to the top (z-direction) and periodic conditions for the others. The
magnitude of the velocity field (arrows) Figs (54) (a) and (b) is found to be maximum
along the edges parallel to the z-direction, essentially because this corresponds to the
largest water-path within the bead array (periodic boundary conditions). Figs (54) (c)
and (d) show the differences induced by the growth of biofilm within the vicinity of
the surface of the bead.
On Fig. (55), we use a numerical configuration rather similar to our experimental
device part II. We solve Stokes equation on images obtained after 10 days of biofilms
growth. Both columns are alimented using the same water supply (identical microorganisms and nutrients thereof). However, the column corresponding to Fig. (55a) has
a flow rate of approximately 0.85 ml/s while the one corresponding to Fig. (55b) is
about 0.05 ml/s. As a consequence of the shear stress induced by the water flow, the
porosity of column (a) is about 0.366 whereas the porosity of column (b) is about
0.240. Using these results, we can also compute the permeability of the two columns,
respectively 1.630 10−5 and 1.437 10−5 m2 for columns (a) and (b). While the relative
difference in permeability is only about 10 %, the pattern of biofilm growth is very
different. In column (a) the growth is relatively uniform, that is, biofilm develops
within area protected from the shear stress and the velocity field is poorly perturbed
by the microbes Fig. (55a). Within column (b) the flow follows preferential pathways
along the column Fig. (55b), that is, the biofilm is growing mainly in the center of the
column. Initially, there is probably higher local velocities nearby the wall because,
177
Figure 54: Calculation of the velocity field (resolution of Stokes equation) using a finite volume
formulation (250x250x250 meshes).
178
2.1 N U M E R I C A L P E R S P E C T I V E S
Figure 55: Calculation of the velocity field (resolution of Stokes equation) using a finite volume
formulation (171x171x300 meshes).
locally, the bead packing is perturbed. Hence, the formation of biofilms might preferentially start in the center of the column, inducing the formation of preferential
pathways for the flow. These results are given as an illustration of the effectiveness of
this kind of calculations, and must be taken extremely precautiously. Generalization
of such behaviors in rich nutrients conditions would require further experiments on
various species and multiple replicates.
Following the same line, one could solve the closure problems developed in Parts III
and IV, to obtain the effective parameters associated with the dispersive macroscale
equations. The local mass equilibrium dispersion tensor as well as the one-equation
non-equilibrium dispersion tensor or two-equations effective parameters can be
solved directly on realistic porous media, and not only on simple 2-D geometries.
2.1.2 Pore-network modeling ?
One caveat when using directly images obtained through X-ray tomography is the
potential size of the data. For example, on Figs. (54) and (55), we subsampled the data
set to obtain a voxel size of 18 µm to compute the results within a reasonable amount
of time. One solution to this issue would be to consider pore-network representation
of the three-dimensional tomography image. A huge amount of image processing
softwares have been developed to undertake this step, that is, to obtain the skeleton
and a pore-network representation (often spheres connected by cylinders, see for
179
180
example [86]). Developing such algorithms for the biofilm problem is rather challenging as one would have to formulate algorithms that would be appropriate for the
resolution of Navier-Stokes equations but also to the dual-phase closure problems.
One could, for example, create two nested spheres and cylinders to delineate the fluid,
the biofilm and the solid. One alternative could also be to develop a fully parallelized
code than can be used to calculate effective parameters on extremely large data sets,
using powerful CPU clusters.
2.1.3 Adaptative macrotransport calculations
Most of the numerical applications of upscaled models to experimental data use one
single formulation that applies for one specific experiment. However, in many cases,
the initial microscale problem that is considered is similar. It would be extremely
interesting to develop codes that can adapt the formulation to the parameters of the
problem. One could, for example, design a program that uses convolutions, threeequation, two-equation or one-equation models on the basis of internal constraints
regarding the physical parameters at play. This would allow the development of one
single code that can be used in, virtually, evey similar problem of mass transport
(under the condition that the microscale formulation is identical).
2.2
E X P E R I M E N TA L P E R S P E C T I V E S
2.2.1 Controlling the microbial species in order to study responses to various environmental stresses
While presenting the imaging work in various conferences, there has been some
interest regarding the specific bacterial species and strains that were used. In our
experiments, we amended water from the Garonne and used this to inoculate our
porous systems. On the one hand this means that we do not control the species
that are present in our system. On the other hand, it also means that our method is
versatile and does not require an a priori staining of the cells. Anyways, working with
specific species is required in order to draw conclusions regarding the behavior of
specific biofilms. Recently, we started collaborations with E. Paul, Y. Pechaud from
the LISBP (INSA Toulouse) and P. Creux, F. Guerton from the University of Pau, to
tackle this issue.
A general view of the experimental device can be found on Fig . (56). E. Coli was
introduced within the porous systems (packed glass beads to have an initial contrast
between the solid phase and the water/biofilm phases) and set at rest for 24 hours.
Then, various flow rates were imposed within the columns using peristaltic pumps as
well as refrigerated (to avoid the formation of other organisms) and aerated (using an
air pump) water mixed with nutrients. The nutrients were introduced in large excess
2.2 E X P E R I M E N TA L P E R S P E C T I V E S
and consisted in a buffered solution of glucose. Oxygen consumption and pH were
also followed at the output of the column.
One single test experiment has been performed and this still represents ongoing
incomplete work. Full results are not available yet, but from observations made with
the naked eye, it seemed that more biofilm has developed in the high flow rate experiment, as opposed to the experiment part II. Three-dimensional imaging has been
done on a Skyscan 1172 (100kV), because glass materials require high energy beams.
Several technical problems need to be tackled, before the reconstructed images can be
used for quantitative analysis. For example, settling was observed within the barium
sulfate suspension as a result of 20 times dilutions (to avoid an acquisition of several
hours). The Micropaque (Guerbet) contains other components in small quantities
that might avoid settling and might be inefficient in low concentrations.
2.2.2 Imaging
The imaging technique as used in this thesis
is not destructive, but still terminal as explained in detail in part II. Hence, it can not
be used to follow the growth of the biomass within one single column. One could
use various replicates, and sacrifice one column every given time step. However,
such experiments would require an additional complexity and this is not necessarily
simple to realize. In addition, following the same column would give more pore-scale
information regarding the growth, even though the averaged parameters should be
similar. Developing a dynamic imaging of biofilms growth within opaque porous
media is still a problem that requires further attention.
DY N A M I C G ROW T H M O N I TO R I N G
While using benchmark tomograph is more accessible
than synchrotron sources, materials such as glass or rock would be easier to image
with high brilliance X-ray sources. In addition, one could use high concentration of
contrast agents and avoid the settling problem. Such synchrotron experiments will
be performed in November 2010, at the Argonne National Laboratory (Chicago) in
collaboration with D. Wildenschild and G. Iltis from the Oregon State University.
SYNCHROTRON SOURCES
2.2.3 Darcy-scale experiments
For the validation and test of the macroscale models as developed in part III and
IV, it would be extremely useful to perform Darcy-scale experiments, in controlled
environments. Such devices would also provide the information that is required in
order to understand biofilm growth and to model it on a larger scale. On Fig (57), one
can see a human-sized column that has been designed during this thesis, and that
will be used for such purposes.
181
182
Figure 56: Experimental device used at the LISBP.
Figure 57: Human sized column filled with expanded polystyrene beads (IMFT)
183
184
2.3
THEORETICAL PERSPECTIVES
2.3.1 Non-linearity
One of the main approximation that has been made in part IV, is to assume a linear
reaction rate within the biofilm matrix. While such an approximation is thoroughly
discussed and has been used countless times, this is mainly a result of our incapacity
to upscale non-linear terms. Developing upscaling techniques that apply to nonlinear reaction rates would be extremely useful. One solution that can be adopted is
to decompose the medium into virtual phases (for example three or four phases for
the biofilm problem).
To understand this approach, we can simply express the reaction rate the following
way
R i (ci ) = R i (hci ii + c̃i ) = R i (hci ii ) + Ṙ i (hci ii )c̃i + O c̃2i
The main problem, is that, in a general case, the terms Ṙ i (hci ii )c̃i + O c̃2i are
non-linear and a closure to the perturbation problem is not straightforward. One has
a solution to this issue when
1. the term R i (ci ) is linear in relation to ci .
2. the term R i (ci ) is non-linear in relation to ci and Ṙ i (hci ii )c̃i + O c̃2i can be
neglected as compared to R i (hci ii ).
To satisfy the proposition number two, we need c̃i as small as possible, to approximate
the reaction rate by its zeroth order Taylor approximation. One way to lower the
pointwise values of c̃i is to create additional virtual phases. The obvious counterpart
is that we filter less information from the microscale and, as a consequence, the
problem will be more complex.
2.3.2 Adaptative macrotransport theory
Usually, the choice of using one-equation, two-equation models is rather intuitive.
For example, even in a single-phase situation, one can consider two-equation models
for momentum Darcy-scale transport, if dead-end pores represent a significative part
of the pore space. In essence, this means that this is not the physical phases that are
important, but rather the topology of the physical processes.
As an alternative to this heuristic fictive separation, one could, for example, solve
numerically the microscale equations or a simpler problem (to be determined) on
several REVs (rather than on the entire system), in order to obtain more information
from the REV, that is, to get a view of the topology of the processes.
2.3 T H E O R E T I C A L P E R S P E C T I V E S
2.3.3 Mobile interfaces and growth/transport coupling
In all the different models presented during this thesis, interfaces are immobile. In
other words, we assume that the characteristic times for biofilm growth is very large
as compared to times associated with transport phenomena. This means that, at any
given time, one can use effective properties that are constant over time (however,
notice that this does not solve the problem of determining laws that apply to the
effective parameters as functions of a larger time scale and this requires additional
considerations).
Even though this approximation is verified in most cases (discussed extensively,
especially in part IV), there certainly arise situations for which both times are of the
same order of magnitude and a coupling would be mandatory. In such cases, it would
be interesting to couple the biofilm motion with the mass transport phenomena, to
understand the dynamics of effective parameters variations. One could, for example,
couple cellular automata with upscaling techniques. Another possibility would be to
image dynamically the porous medium with biofilm. One other way around would
be to devise a downscaling technique that can recover the pore-scale geometry of the
biofilm, in the condition that, say, the mass exchange coefficient is maximized and
the averaged shear stress is minimized. One interest of the last proposition is that
one would be able to identify what are the macroscopic parameters that drive biofilm
growth.
2.3.4 And the other scales ?
In this work, we are particularly interested in upscaling from the pore-scale/biofilmscale to the Darcy-scale, albeit we generalize the upscaling process in part III, in
the case of non-reactive solute transport. As previously discussed, one of the main
assumption of our developments is the form of the microscale problem that is used.
Upscaling using volume averaging with closure necessarily requires that the boundary
value problems that describes the phenomena of interest are well described at a given
scale. One could start, for example, on a cellular or even on molecular-scale. The
convection-diffusion problem considered in this thesis is known to be an excellent
description of the processes as there has been substantial experimental support for
this approximation.
However, there are many other scales that require further attention. For example,
one could consider upscaling from the pore-scale to the Darcy-scale in a dual-phase
configuration and then to a larger heterogeneous dual-region medium, to obtain fourequation models. One other interesting area of research concerns channeled biofilms.
There has been some work regarding this physical problem [11]. However, this requires further investigation, and calculating effective parameters of biofilms on the
basis of direct 3-D images of the channels, using for example X-ray microtomography,
would considerably help in apprehending their function within biofilms.
185
Part VI
R E F E R E E D P U B L I C AT I O N S B Y T H E AU T H O R
Author's personal copy
Advances in Water Resources 33 (2010) 1075–1093
Contents lists available at ScienceDirect
Advances in Water Resources
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a d v wa t r e s
Modeling non-equilibrium mass transport in biologically reactive porous media
Yohan Davit a,c,⁎, Gérald Debenest a,b, Brian D. Wood d, Michel Quintard a,b
a
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse) Allée Camille Soula F-31400 Toulouse, France
CNRS; IMFT F-31400 Toulouse, France
c
Université de Toulouse; INPT, UPS; ECOLAB Rue Jeanne Marvig F-31055 Toulouse, France
d
School of Chemical, Biological, and Environmental Engineering, Oregon State University, Corvallis, OR 97331, United States
b
a r t i c l e
i n f o
Article history:
Received 22 September 2009
Received in revised form 20 June 2010
Accepted 22 June 2010
Available online 30 June 2010
Keywords:
Porous media
Biofilms
Upscaling
Volume averaging
Non-equilibrium
One-equation model
a b s t r a c t
We develop a one-equation non-equilibrium model to describe the Darcy-scale transport of a solute
undergoing biodegradation in porous media. Most of the mathematical models that describe the macroscale
transport in such systems have been developed intuitively on the basis of simple conceptual schemes. There
are two problems with such a heuristic analysis. First, it is unclear how much information these models are
able to capture; that is, it is not clear what the model's domain of validity is. Second, there is no obvious
connection between the macroscale effective parameters and the microscopic processes and parameters. As
an alternative, a number of upscaling techniques have been developed to derive the appropriate macroscale
equations that are used to describe mass transport and reactions in multiphase media. These approaches
have been adapted to the problem of biodegradation in porous media with biofilms, but most of the work has
focused on systems that are restricted to small concentration gradients at the microscale. This assumption,
referred to as the local mass equilibrium approximation, generally has constraints that are overly restrictive.
In this article, we devise a model that does not require the assumption of local mass equilibrium to be valid.
In this approach, one instead requires only that, at sufficiently long times, anomalous behaviors of the third
and higher spatial moments can be neglected; this, in turn, implies that the macroscopic model is well
represented by a convection–dispersion–reaction type equation. This strategy is very much in the spirit of
the developments for Taylor dispersion presented by Aris (1956). On the basis of our numerical results, we
carefully describe the domain of validity of the model and show that the time-asymptotic constraint may be
adhered to even for systems that are not at local mass equilibrium.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction
1.1. One-equation local mass equilibrium model
Biodegradation in porous media has been the subject of extensive
studies from the environmental engineering point of view [1–5].
Reactions are mediated by microorganisms (primarily bacteria, fungi,
archaea, and protists, although others may be present) aggregated and
coated within an extracellular polymeric matrix; together, these
which form are generically called biofilms. There has been significant
interest for their role in bioremediation of soils and subsurfaces [6–12]
and, more recently, for their application to supercritical CO2 storage
[13,14]. Numerous models for describing the transport of solutes, such
as organic contaminants or injected nutrients, through geological
formations as illustrated in Fig. 1, have been developed. Reviews of
these mathematical and physical representations of biofilms processes
can be found in [15] and [16].
In many applications, the macroscopic balance laws for mass
transport in such hierarchical porous media with biofilms have been
elaborated by inspection. For example, the advection–dispersion–
reaction type Eq. (1) is commonly considered to describe the Darcyscale transport
γ of a contaminant/nutrient represented by a concentration cγ in the water γ-phase. Brackets notations are here as a
reminder that this concentration must be defined in some averaged
sense.
⁎ Corresponding author. Université de Toulouse; INPT, UPS; IMFT (Institut de
Mécanique des Fluides de Toulouse) Allée Camille Soula F-31400 Toulouse, France.
E-mail addresses: [email protected] (Y. Davit), [email protected] (G. Debenest),
[email protected] (B.D. Wood), [email protected] (M. Quintard).
0309-1708/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.advwatres.2010.06.013
D Eγ
∂ cγ
∂t
D Eγ D Eγ
D Eγ +R
+ vγ ⋅∇ cγ = ∇⋅ D⋅∇ cγ
ð1Þ
γ
In this expression, vγ is the groundwater velocity and D is a
dispersion tensor. The reaction
rate R is usually assumed to have a
γ
hcγ i
, where α and K are parameters
Monod form R = −α
γ
hcγ i + K
(discussed in Section 3.5). It is common to assume that the solute
transport can be uncoupled from the growth process [17,18], that is,
to consider that the characteristic times for these two processes are
International Journal of Heat and Mass Transfer 53 (2010) 4985–4993
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Equivalence between volume averaging and moments matching techniques for
mass transport models in porous media
Yohan Davit a, Michel Quintard a,b, Gérald Debenest a,b,c,*
a
Université de Toulouse, INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France
CNRS, IMFT, F-31400 Toulouse, France
c
Ecolab UMR 5245 (INP-ENSAT, UPS, CNRS), Equipe ECOGEN, Av. de l’Agrobiopole, BP 32607 Auzeville Tolosane, 31326 Castanet Tolosan Cedex, France
b
a r t i c l e
i n f o
Article history:
Received 13 July 2009
Received in revised form 3 March 2010
Accepted 9 April 2010
Available online 14 June 2010
Keywords:
Porous media
Non-equilibrium
Two-equation model
Moments
Time-asymptotic
Volume averaging
a b s t r a c t
This paper deals with local non-equilibrium models for mass transport in dual-phase and dual-region
porous media. The first contribution of this study is to formally prove that the time-asymptotic moments
matching method applied to two-equation models is equivalent to a fundamental deterministic perturbation decomposition proposed in Quintard et al. (2001) [1] for mass transport and in Moyne et al. (2000)
[2] for heat transfer. Both theories lead to the same one-equation local non-equilibrium model. It has very
broad practical and theoretical implications because (1) these models are widely employed in hydrology
and chemical engineering and (2) it indicates that the concepts of volume averaging with closure and of
matching spatial moments are equivalent in the one-equation non-equilibrium case. This work also aims
to clarify the approximations that are made during the upscaling process by establishing the domains of
validity of each model, for the mobile–immobile situation, using both a fundamental analysis and numerical simulations. In particular, it is demonstrated, once again, that the local mass equilibrium assumptions
must be used very carefully.
Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction
In this paper, we investigate the behavior of widely used models, especially in subsurface hydrology, for describing the transport
of a tracer through dual-phase and dual-region porous media similar to those presented in Fig. 1. At the microscopic scale (porescale or Darcy-scale in this article), these two systems are usually
thought as two continua, in which advection and Fickian diffusion/
dispersion are dominant. Upscaling mass balanced equations from
the pore-scale to the Darcy-scale in a dual-phase porous medium
is, mathematically speaking, equivalent to upscaling from the
Darcy-scale to the large-scale in a dual-region porous medium.
Hence, in this work, we consider a general framework defined by
the mathematical structure of the boundary-value problem at the
small scale rather than by the scale itself and the corresponding
physical phenomena.
A general solution to the dispersion problem at the macroscopic
scale exhibits time and spatial convolutions (non-locality) [3–8].
Direct Numerical Simulations can be used to get an accurate
response, but it is not often tractable for most of porous systems
because of the complexity involved. Degraded models, but still
rather accurate if one is concerned with the estimate of the flux exchanged between the different phases/regions, can be derived under the form of mixed models for mobile/immobile systems, i.e.
systems with high diffusivity contrast and no advection in the
low diffusivity phase/region [9–12]. In many practical applications,
relevant constraints can be formulated concerning characteristic
times and length scales [13], that is, when the different length
scales of the system are separated by several orders of magnitude,
non-local effects tend to disappear. One way of describing the
transport while conserving partly the convolutions effects consists
in using a two-equation quasi-stationary model (Eqs. (1) and (2))
involving two macroscopic concentrations, one for each phase/region c and x. Herein, the bracket notation is used as a reminder
that concentrations appearing in the macroscopic equations are
defined in a volume averaged sense. ei is the volume fraction of
the i-phase/region and the star notation refers to the model effective parameters. In particular, a** is a first order mass exchange
coefficient.
Two-equation model:
n
* Corresponding author at: Université de Toulouse, INPT, UPS, IMFT (Institut de
Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France.
E-mail addresses: [email protected] (Y. Davit), [email protected] (M. Quintard),
[email protected] (G. Debenest).
0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2010.05.032
c
x
c
ec @ t hcc ic þ V
cc $hcc i þ Vcx $hcx i ¼ $ Dcc $hcc i
n
o
x
þ $ D
a hcc ic hcx ix
cx $hcx i
o
ð1Þ
Journal of Microscopy, 2010
doi: 10.1111/j.1365-2818.2010.03432.x
Received 29 October 2009, accepted 14 June 2010
Imaging biofilm in porous media using X-ray computed
microtomography
Y . D A V I T ∗ ‡, G . I L T I S ||, G . D E B E N E S T ∗, †,
S . V E R A N - T I S S O I R E S ∗ , D . W I L D E N S C H I L D ||, M . G E R I N O ‡, §
&∗ M . Q U I N T A R D ∗, †
Université de Toulouse; INPT, UPS; IMFT(Institut de Mécanique des Fluides de Toulouse), Allée
Camille Soula Toulouse, France
†CNRS; IMFT F-31400 Toulouse, France
‡Université de Toulouse; INPT, UPS; ECOLAB Rue Jeanne Marvig F-31055 Toulouse, France
§CNRS; ECOLAB F-31055 Toulouse, France
||School of Chemical, Biological, and Environmental Engineering, Oregon State University Corvallis,
Oregon, U.S.A.
Key words: Biofilm, imaging, porous media, X-ray tomography.
Summary
In this study, a new technique for three-dimensional imaging
of biofilm within porous media using X-ray computed
microtomography is presented. Due to the similarity in
X-ray absorption coefficients for the porous media (plastic),
biofilm and aqueous phase, an X-ray contrast agent is required
to image biofilm within the experimental matrix using
X-ray computed tomography. The presented technique utilizes
a medical suspension of barium sulphate to differentiate
between the aqueous phase and the biofilm. Potassium
iodide is added to the suspension to aid in delineation
between the biofilm and the experimental porous medium.
The iodide readily diffuses into the biofilm while the barium
sulphate suspension remains in the aqueous phase. This allows
for effective differentiation of the three phases within the
experimental systems utilized in this study. The behaviour of
the two contrast agents, in particular of the barium sulphate,
is addressed by comparing two-dimensional images of biofilm
within a pore network obtained by (1) optical visualization and
(2) X-ray absorption radiography. We show that the contrast
mixture provides contrast between the biofilm, the aqueousphase and the solid-phase (beads). The imaging method is
then applied to two three-dimensional packed-bead columns
within which biofilm was grown. Examples of reconstructed
images are provided to illustrate the effectiveness of the
method. Limitations and applications of the technique are
discussed. A key benefit, associated with the presented method,
is that it captures a substantial amount of information
Correspondence to: Yohan Davit, Université de Toulouse; INPT, UPS; IMFT (Institut
de Mécanique des Fluides de Toulouse), Allée Camille Soula F-31400 Toulouse,
France. Tel: +33-5-3432-2882; fax: +33-5-3432-2993; e-mail: [email protected]
C 2010 The Author
C 2010 The Royal Microscopical Society
Journal of Microscopy regarding the topology of the pore-scale transport processes.
For example, the quantification of changes in porous media
effective parameters, such as dispersion or permeability,
induced by biofilm growth, is possible using specific upscaling
techniques and numerical analysis. We emphasize that the
results presented here serve as a first test of this novel approach;
issues with accurate segmentation of the images, optimal
concentrations of contrast agents and the potential need for
use of synchrotron radiation sources need to be addressed
before the method can be used for precise quantitative analysis
of biofilm geometry in porous media.
Introduction
Microorganisms (primarily bacteria, fungi and algae), in wet
or aqueous environments, tend to aggregate and grow on
surfaces, embedded within extracellular polymeric substances
(EPS) (Costerton et al., 1995; Sutherland 2001). These sessile
communities, termed biofilms, are ubiquitous in industry
(Ganesh Kumar & Anand 1998), in medicine and natural
environments (Hall-Stoodley et al., 2004). Biofilm cells, when
compared with planktonic cells, have been documented
to be more resistant to antibiotics and biocides (Costerton
et al., 1999; Stewart, 2001; Davies, 2003; Hall-Stoodley
et al., 2004). Hence, the development of biofilms can have
undesirable and potentially harmful consequences in medical
applications (Diosi et al., 2003; Lee & Kim, 2003), but
can also be useful in natural or engineered systems such
as wastewater treatment processes (Lazarova & Manem,
2000), bioremediation (Rittmann et al., 2000) or CO2 storage
(Mitchell et al., 2009). In medical, natural, as well as
engineered systems, biofilm control strategies, based on a
better understanding of biofilm growth characteristics as well
September 2008
EPL, 83 (2008) 64001
doi: 10.1209/0295-5075/83/64001
www.epljournal.org
Intriguing viscosity effects in confined suspensions:
A numerical study
Y. Davit and P. Peyla(a)
Laboratoire de Spectrométrie Physique, Université Joseph Fourier - Grenoble 1, BP87,
F-38402 Saint Martin d’Hères, France, EU
received 6 May 2008; accepted in final form 4 August 2008
published online 9 September 2008
PACS
PACS
PACS
47.57.E- – Suspensions
47.57.Qk – Rheological aspects
47.11.-j – Computational methods in fluid dynamics
Abstract – The effective viscosity of dilute and semi-dilute suspensions in a shear flow in a
microfluidic configuration is studied numerically. The suspension is composed of monodisperse
and non-Brownian hard spherical buoyant particles confined between two walls in a shear flow.
An abrupt change of the viscosity behaviour occurs with strong confinements: when the wallto-wall distance is below five times the radius of the particles, we obtain a change of the sign
of the contribution of the hydrodynamic interactions to the effective viscosity. This effect is
the macroscopic counterpart of the peculiar micro-hydrodynamics of confined suspensions due
to the influence of walls. In addition, for higher concentrations (above 25%), we find that the
viscosity meets a minimum when the inter-wall distance is around five times the sphere radius. This
phenomenon is reminiscent of the Fahraeus-Lindqvist effect for blood confined in small capillaries.
However, we show that for sheared confined semi-dilute suspensions, the physical origin of this
minimum is not due to a migration effect but to the change of hydrodynamic interactions.
c EPLA, 2008
Copyright Introduction. – Solid particles suspended in a conventional liquid form a suspension. This composite fluid
constitutes a widespread fluid material in nature as well
as in industry [1]. The particles can be spatially confined
in porous media, in biological capillaries or in microfluidic devices [2]. In these situations, rheological phenomena due to interactions between device boundaries and
fluid constituents become much more important than in
conventional cases. These interactions of hydrodynamic
origin play a crucial role, particularly, they appear to affect
notably the effective viscosity ηef f of confined sheared
suspensions as shown in this letter. Of course, considering microfluidics devices, pressure-driven flows are very
important. However, in order to understand the role of
the walls on hydrodynamic interactions, it is preferable to
study a suspension in a simple shear geometry in order to
estimate its effective viscosity. In addition, it should allow
rheologists to easily compare their experimental results
with our numerical predictions.
It is well known that the relative viscosity of a semidilute sheared suspension follows a virial expansion:
(a) E-mail:
[email protected]
∆η ηef f − η0
=
= [η]1 φ + [η]2 φ2 + O(φ3 ),
η0
η0
(1)
where φ is the volume fraction defined as the volume of the
particles normalized to the total volume of the suspension
and η0 is the viscosity of the fluid. For a non-confined
suspension of hard spheres, the linear term (the intrinsic
viscosity) is [η]1,∞ = 2.5 as calculated by Einstein [3]
for a strong dilution (i.e. particles are far enough not
to interact with each other through hydrodynamics).
Subscript ∞ indicates that we refer to non-confined
suspensions. When increasing φ, the semi-dilute regime
is reached, the particles get closer than in the dilute
regime and start to interact hydrodynamically. Batchelor
and Green [4] showed that the hydrodynamic interactions
contribute to the second order in φ. They found [η]2,∞ =
5.2 ± 0.3 for a non-confined and non-Brownian suspension
where the particles are uniformly distributed. Since then,
a more precise estimation of [η]2,∞ = 5.0 has been achieved
by Cichoki and Felderhof [5].
We find that when the wall-to-wall distance (or gap)
w decreases, the linear term [η]1 (dominant for strongly
dilute cases, i.e. φ 1) increases. This is due to dissipation which is enhanced for smaller gaps. However,
64001-p1
Part VII
APPENDIX
A
M AT H E M AT I C A L D E V E L O P M E N T F O R T H E T W O - E Q U AT I O N
NON-REACTIVE MODEL
In this Appendix, we develop the volume averaged equations for each phase. We start
with the microscale description of the medium associated with Fig (58).
∂cγ
+ ∇ · (vγ cγ ) = ∇ · (
∂t
γ − phase :
− (nγσ ·
BC1 :
BC2 :
BC3 :
BC4 :
Dγ · ∇cγ)
Dγ) · ∇cγ = 0
cω = cγ
Dγ) · ∇cγ = − (nγω · Dω) · ∇cω
− (nωσ · Dω ) · ∇cω = 0
− (nγω ·
ω − phase :
∂cω
+ ∇ · (vω cω ) = ∇ · (
∂t
(A.1)
on S γσ
(A.2a)
on S γω
(A.2b)
on S γω
(A.2c)
on S ωσ
(A.2d)
Dω · ∇cω)
(A.3)
To develop equations governing mass transport at the macroscopic scale, we need
to average each equation
γ − phase :
ω − phase :
Dγ · ∇cγ)i
Dω · ∇cω)i
∂cγ
i + h∇ · (vγ cγ )i = h∇ · (
∂t
∂cω
h
i + h∇ · (vω cω )i = h∇ · (
∂t
h
(A.4)
(A.5)
Then, we are confronted to the classical problem of averaging time derivatives and
spatial operators. For this purpose, we use the following theorems with w the velocity
195
196
M AT H E M AT I C A L D E V E L O P M E N T F O R T H E T W O - E Q U AT I O N N O N - R E A C T I V E M O D E L
Figure 58: Pore-scale description of a Darcy-scale averaging volume
of the interface.
General transport theorem [273]
∂hcγ i 1
∂cγ
i=
−
h
∂t
∂t
V
∂cω
∂hcω i 1
h
i=
−
∂t
∂t
V
Z
Z
(nγω · w) cγ dS
(A.6)
(nωγ · w) cω dS
(A.7)
S γω (t)
S ωγ (t)
Spatial Averaging theorems [129, 119]
Z
Z
1
1
nγω cγ dS +
nγσ cγ dS
h∇cγ i = ∇hcγ i +
V Sγω
V Sγσ
Z
Z
1
1
h∇cω i = ∇hcω i +
nωγ cω dS +
nωσ cω dS
V Sωγ
V Sωσ
Imposing w = 0 leads to
(A.8)
(A.9)
∂εγ hcγ iγ
+ ∇ · (εγ hcω iγ hvγ iγ ) =
∂t
D
∇
· { γ · ∇ {εγ hcγ iγ } +
+
1
V
−
∇ · hc̃γ vγ i
∂εω hcω iω
+ ∇ · (εω hcω iω hvω iω ) =
∂t
∇
+
−
Z
1
V
Z
S γω
nγω cγ dS +
D
1
(nγω · γ ) · ∇cγ dS +
V
S γω
Z
S γσ
1
V
Z
!
S γσ
nγσ cγ dS }
D
(nγσ · γ ) · ∇cγ dS
(A.10)
!
Z
1
· { ω · ∇ {εω hcω
nωγ cω dS +
nωσ cω dS }
V S ωγ
V S ωσ
Z
Z
1
1
(nωγ · ω ) · ∇cω dS +
V S ωγ
V S ωσ
D
1
iω } +
Z
D
D
∇ · hc̃ω vω i
(A.11)
We use the following decompositions cγ = hcγ iγ + c̃γ , cω
ω
ω
hcω iω
x+y = hcω ix + y · ∇hcω ix + · · ·
= hcω iω + c̃ω
and
(A.12)
hcγ iγx+y = hcγ iγx + y · ∇hcγ iγx + · · ·
(A.13)
where x is the vector pointing the position of the center of the REV and y is the vector
pointing inside the REV. Then we can neglect all the non-local terms involving y
provided that R20 L2 [274], where L is a characteristic field-scale length, and this is
expressed by
1
V
Z
S γω
nγω cγ dS +
1
V
Z
S γσ
nγσ cγ dS
=
1
V
+
Z
γ
S γω
1
V
nγω hcγ ix+y dS +
Z
nγω c̃γ dS +
S γω
1
V
Z
1
V
Z
1
V
Z
γ
S γσ
S γσ
nγσ hcγ ix+y dS
nγσ c̃γ dS
!
Z
1
' hcγ ix
nγω dS +
nσω dS
V S σω
S γω
Z
Z
1
1
+
nγω c̃γ dS +
nγσ c̃γ dS
(A.14)
V S γω
V S γσ
Z
Z
Z
Z
1
1
1
1
nωγ cω dS +
nωσ cω dS =
nωγ hcω iω
nωσ hcω iω
x+y dS +
x+y dS
V S ωγ
V S ωσ
V S ωγ
V S ωσ
Z
Z
1
1
+
nωγ c̃ω dS +
nωσ c̃ω dS
V S ωγ
V S ωσ
!
Z
Z
1
1
' hcω iω
n
dS
+
n
dS
ωγ
ωσ
x
V S ωγ
V S ωσ
Z
Z
1
1
+
nωγ c̃ω dS +
nωσ c̃ω dS
(A.15)
V S ωγ
V S ωσ
γ
Then, using spatial averaging theorems for the phase indicators gives
Z
Z
1
1
− ∇εω =
nωγ dS +
nωσ dS
V Sωγ
V Sωσ
Z
Z
1
1
−∇εγ =
nγω dS +
nγσ dS
V Sγω
V Sγσ
(A.16)
(A.17)
Hence, we have
1
V
Z
S ωγ
nωγ cω dS +
1
V
Z
S ωσ
197
nωσ cω dS ' −∇εω hcω iω
x +
1
V
Z
S ωγ
nωγ c̃ω dS +
1
V
Z
S ωσ
nωσ c̃ω dS (A.18)
198
1
V
Z
S γω
nγω cγ dS +
1
V
Z
S γσ
γ
nγσ cγ dS ' −∇εγ hcγ ix +
1
V
Z
S γω
nγω c̃γ dS +
1
V
Z
S σω
nσω c̃γ dS (A.19)
Finally, injecting Eqs (B.18) and (B.19) into (B.10) and (B.11) and doing the same
approximations for the macroscopic fluxes leads to
∂εγ hcγ iγ
+ ∇ · (εγ hciγ hvγ iγ ) =
∂t
D
∇
· {εγ γ · ∇hcγ iγ +
+
1
V
−
∇ · hc̃γ vγ i
∂εω hcω iω
+ ∇ · (εω hcω iω hvω iω ) =
∂t
D
∇
Z
1
Vγ
Z
S γω
nγω c̃γ dS +
D
1
(nγω · γ ) · ∇c̃γ dS +
V
S γω
S γσ
Z
!
S γσ
nγσ c̃γ dS }
D
(nγσ · γ ) · ∇c̃γ dS
(A.20)
Z
· {εω ω ·
Z
+
1
V
−
∇ · hc̃ω vω i
S ωγ
Z
1
Vγ
Z
!
1
1
∇hcω iω +
nωγ c̃ω dS +
nωσ c̃ω dS }
Vω S ωγ
Vω S ωσ
D
(nωγ · ω ) · ∇c̃ω dS +
1
V
Z
S ωσ
D
(nωσ · ω ) · ∇c̃ω dS
(A.21)
B
M AT H E M AT I C A L D E V E L O P M E N T F O R T H E T W O - E Q U AT I O N
NON-CLOSED REACTIVE MODEL
In this Appendix, we develop the volume averaged equations for each phase. We start
with the pore-scale description of the medium
∂cγ
+ ∇ · (cγ vγ ) = ∇ · (
∂t
γ − phase :
BC1 :
BC2 :
BC3 :
BC4 :
− (nγσ ·
Dγ · ∇cγ)
Dγ) · ∇cγ = 0
cω = cγ
− (nγω · γ ) · ∇cγ = − (nγω ·
− (nωσ · ω ) · ∇cω = 0
D
D
Dω) · ∇cω
∂cω
= ∇·(
∂t
ω − phase :
(B.1)
on S γσ
(B.2a)
on S γω
(B.2b)
on S γω
(B.2c)
on S ωσ
(B.2d)
Dω · ∇cω) + R ω
(B.3)
To develop equations governing mass transport at the macroscopic scale, we need to
average each equation
h
γ − phase :
ω − phase :
D
∂cγ
i + h∇ · (cγ vγ )i = h∇ · ( γ · ∇cγ )i
∂t
∂cω
h
i = h∇ · ( ω · ∇cω )i + hR ω i
∂t
D
(B.4)
(B.5)
Then, we are confronted to the classical problem of averaging time derivatives and
spatial operators. For this purpose, we use the following theorems
General transport theorem [273]
∂cγ
∂hcγ i 1
h
i=
−
∂t
∂t
V
∂cω
∂hcω i 1
i=
−
h
∂t
∂t
V
Z
Z
(nγω · w) cγ dS
(B.6)
(nωγ · w) cω dS
(B.7)
S γω (t)
S ωγ (t)
199
200
M AT H E M AT I C A L D E V E L O P M E N T F O R T H E T W O - E Q U AT I O N N O N - C L O S E D R E A C T I V E M O D E L
with w the velocity of the interface.
Spatial Averaging theorems [129, 119]
Z
Z
1
1
nγω cγ dS +
nγσ cγ dS
h∇cγ i = ∇hcγ i +
V Sγω
V Sγσ
Z
Z
1
1
h∇cω i = ∇hcω i +
nωγ cω dS +
nωσ cω dS
V Sωγ
V Sωσ
(B.8)
(B.9)
It has already been emphasized [285] that characteristics times associated to
biofilm motion are long compared to the one associated with the mass transport so
that we can write
∂εγ hcγ iγ
+ ∇ · (εγ hciγω hvγ iγ ) =
∂t
∇
+
−
∂εω hcω iω
=
∂t
∇
!
Z
1
·{ γ ·
nγω cγ dS +
nγσ cγ dS }
V S γω
V S γσ
Z
Z
1
1
(nγω · γ ) · ∇cγ dS +
V S γω
V S γσ
D
1
∇ {εγ hcγ iγ } +
D
D
∇.hĉγ vγ i
D
· { ω · ∇ {εω hcω iω } +
1
+
V
Z
Z
(B.10)
1
V
Z
S ωγ
nωγ cω dS +
D
1
(nωγ · ω ) · ∇cω dS +
V
S ωγ
Z
S ωσ
Z
1
V
!
S ωσ
nωσ cω dS }
D
+εω hR ω iω
(B.11)
We use the following decompositions cγ = hcγ iγ + c̃γ , cω = hcω iω + c̃ω and
ω
ω
hcω iω
x+y = hcω ix + y · ∇hcω ix + · · ·
hcγ iγx+y
γ
(B.12)
γ
= hcγ ix + y · ∇hcγ ix + · · ·
(B.13)
where x is the vector pointing the position of the center of the REV and y is the vector
pointing inside the REV. Then we can neglect all the non-local terms involving y
provided that R20 L2 [274], where L is a characteristic field-scale length, and this is
expressed by
1
V
Z
S γω
nγω cγ dS +
1
V
Z
S γσ
nγσ cγ dS
=
1
V
+
'
Z
γ
S γω
1
V
nγω hcγ ix+y dS +
Z
S γω
nγω c̃γ dS +
Z
Z
γ
S γσ
S γσ
nγσ hcγ ix+y dS
nγσ c̃γ dS
!
Z
1
hcγ ix
nγω dS +
nσω dS
V S γσ
S γω
Z
Z
1
1
+
nγω c̃γ dS +
nγσ c̃γ dS
V S γω
V S γσ
γ
1
V
Z
1
V
1
V
(B.14)
M AT H E M AT I C A L D E V E L O P M E N T F O R T H E T W O - E Q U AT I O N N O N - C L O S E D R E A C T I V E M O D E L
1
V
Z
S ωγ
nωγ cω dS +
Z
1
V
S ωσ
nωσ cω dS
=
1
V
+
'
Z
S ωγ
1
V
nωγ hcω iω
x+y dS +
Z
S ωγ
Z
1
V
Z
S ωγ
nωγ dS +
1
nωγ c̃ω dS +
V
S ωγ
Z
S ωσ
S ωσ
Z
1
V
hcω iω
x
1
+
V
nωγ c̃ω dS +
1
V
Z
nωσ hcω iω
x+y dS
nωσ c̃ω dS
Z
1
V
!
S ωσ
S ωσ
nωσ dS
nωσ c̃ω dS
(B.15)
Then, using spatial averaging theorems for unity gives
R
− ∇εω = V1
−∇εγ
R
1
S ωγ nωγ dS + V
R
1
=V
S ωσ nωσ dS
(B.16)
S γσ nγσ dS
(B.17)
R
1
S γω nγω dS + V
Hence, we have
1
V
Z
S ωγ
1
V
nωγ cω dS +
1
V
Z
S ωσ
Z
S γω
nωσ cω dS ' −∇εω hcω iω
x +
nγω cγ dS +
1
V
Z
γ
S γσ
Z
1
V
nγσ cγ dS ' −∇εγ hcγ ix +
S ωγ
1
V
nωγ c̃ω dS +
1
V
Z
S γω
nγω c̃γ dS +
Z
1
V
S ωσ
nωσ c̃ω dS (B.18)
Z
S σω
nσω c̃γ dS
(B.19)
Finally, injecting Eqs (B.18) and (B.19) into (B.10) and (B.11) leads to
∂εγ hcγ iγ
∂t
∂εω hcω iω
=
∂t
∇
∇
D
· {εγ γ ·
Z
+
1
V
−
∇.hĉγ vγ i
S γω
Z
D
(nγω · γ ) · ∇cγ dS +
1
V
Z
D
Z
+
1
V
+
εω hR ω iω
!
S γσ
D
(B.20)
Z
· {εω ω ·
S ωγ
Z
1
1
∇hcγ iγ +
nγω c̃γ dS +
nγσ c̃γ dS }
Vγ S γω
Vγ S γσ
Z
!
1
1
∇hcω iω +
nωγ c̃ω dS +
nωσ c̃ω dS }
Vω S ωγ
Vω S ωσ
D
(nωγ · ω ) · ∇cω dS +
1
V
Z
S ωσ
D
(B.21)
201
C
M AT H E M AT I C A L D E V E L O P M E N T F O R T H E O N E - E Q U AT I O N
PECULIAR DECOMPOSITION MODEL
In this part, we develop the macroscopic one-equation non-closed form of the model
starting with the averaged equations for each phase
∂εγ hcγ iγ
∂t
∂εω hcω iω
=
∂t
D
∇
· {εγ γ · ∇hcγ iγ +
+
1
V
−
∇.hĉγ vγ i
Z
Z
S γω
nγω c̃γ dS +
D
1
(nγω · γ ) · ∇cγ dS +
V
S γω
D
∇
· {εω ω · ∇hcω iω +
+
1
V
+
εω hR ω iω
Z
1
Vγ
D
1
Vω
Z
1
Vγ
Z
S γσ
!
nγσ c̃γ dS }
D
S γσ
(C.1)
Z
S ωγ
nωγ c̃ω dS +
1
(nωγ · ω ) · ∇cω dS +
V
S ωγ
Z
S ωσ
1
Vω
Z
S ωσ
!
nωσ c̃ω dS }
D
(C.2)
Then, we make the flux term disappear by summing equations over the γ − phase
and the ω − phase and by using the flux-continuity hypothesis at the interface
between γ and ω.
∂hci
∂t
∇
+
+
+
D
D
· {εω ω · ∇hcω iω + εγ γ · ∇hcγ iγ }
!
Z
Z
1
1
∇·{ ω ·
nωγ c̃ω dS +
nωσ c̃ω dS }
V S ωγ
V S ωσ
!
Z
Z
1
1
∇·{ γ ·
nγω c̃γ dS +
nγσ c̃γ dS }
V S γω
V S γσ
D
D
εω hR ω iω − ∇.hĉγ vγ i
(C.3)
203
204
M AT H E M AT I C A L D E V E L O P M E N T F O R T H E O N E - E Q U AT I O N P E C U L I A R D E C O M P O S I T I O N M O D E L
Then, we make intrinsic total average equation appear dividing by εγ + εω (supposed constant over time and space) as it is the one used in the decompositions of
concentrations.
∂hciγω
+∇·
∂t
εγ
hciγω hvγ iγ =
εγ + ε ω
εω
εγ + ε ω
Dω · ∇hcω iω + ε
εγ
Dγ · ∇hcγ iγ }
∇
·{
+
∇·{
εω
εγ + ε ω
Dω ·
+
∇·{
εγ
εγ + ε ω
Dγ ·
+
1
εω
hR ω iω −
∇.hĉγ vγ i
εγ + ε ω
εγ + εω
γ + εω
!
Z
Z
1
1
nωγ ĉω dS +
nωσ ĉω dS }
Vω S ωγ
Vω S ωσ
!
Z
Z
1
1
nγω ĉγ dS +
nγσ ĉγ dS }
Vγ S γω
Vγ S γσ
(C.4)
Finally, we use the following relations
hcγ iγ = hciγω + hĉγ iγ
(C.5)
c̃γ = ĉγ − hĉγ iγ
(C.7)
hcω i
ω
= hci
γω
+ hĉω i
c̃ω = ĉω − hĉω i
ω
(C.6)
ω
(C.8)
which gives
∂hciγω
+∇·
∂t
εγ
hciγω hvγ iγ =
εγ + ε ω
∇
+
+
+
−
−
+
D
D
εγ
εω
γω }
ω+
γ · ∇hci
εγ + ε ω
εγ + ε ω
εγ
εω
ω
γ
∇·{
ω · ∇hĉω i +
γ · ∇hĉγ i }
εγ + ε ω
εγ + εω
!
Z
Z
1
εω
1
∇·{
nωγ ĉω dS +
nωσ ĉω dS }
ω·
εγ + ε ω
Vω S ωγ
Vω S ωσ
!
Z
Z
εγ
1
1
∇·{
nγω ĉγ dS +
nγσ ĉγ dS }
γ·
εγ + ε ω
Vγ S γω
Vγ S γσ
!
Z
Z
1
1
εω
ω
ω
∇·{
nωγ hĉω i dS +
nωσ hĉω i dS }
ω·
εγ + ε ω
Vω S ωγ
Vω S ωσ
!
Z
Z
εγ
1
1
γ
γ
∇·{
nγω hĉγ i dS +
nγσ hĉγ i dS }
γ·
εγ + ε ω
Vγ S γω
Vγ S γσ
·{
D
D
D
D
D
D
1
εω
hR ω iω −
∇.hĉγ vγ i
εγ + ε ω
εγ + εω
(C.9)
Then, using spatial averaging theorems for unity gives
Z
Z
1
1
− ∇εω =
nωγ dS +
nωσ dS
V Sωγ
V Sωσ
Z
Z
1
1
−∇εγ =
nγω dS +
nγσ dS
V S γω
V S γσ
(C.10)
M AT H E M AT I C A L D E V E L O P M E N T F O R T H E O N E - E Q U AT I O N P E C U L I A R D E C O M P O S I T I O N M O D E L
So that we can write
∂hciγω
+∇·
∂t
εγ
hciγω hvγ iγ =
ε γ + εω
∇
+
+
+
+
+
+
D
D
εγ
εω
γω }
ω+
γ · ∇hci
εγ + ε ω
ε γ + εω
εγ
εω
ω
γ
∇·
ω · ∇hĉω i +
γ ∇hĉγ i
εγ + εω
εγ + εω
!
Z
Z
εω
1
1
}
∇·{
·
n
ĉ
dS
+
n
ĉ
dS
ω
ωγ ω
ωσ ω
εγ + ε ω
Vω S ωγ
Vω S ωσ
!
Z
Z
εγ
1
1
∇·{
nγω ĉγ dS +
nγσ ĉγ dS }
γ·
εγ + ε ω
Vγ S γω
Vγ S γσ
1
ω
∇·
ω · ∇εω hĉω i
εγ + εω
1
γ
∇·
γ · ∇εγ hĉγ i
εγ + εω
·{
D
D
D
D
D
D
1
εω
hR ω iω −
∇.hĉγ vγ i
ε γ + εω
εγ + εω
(C.11)
205
D
INTRODUCTION ET CONCLUSION EN FRANÇAIS
D.1
CONTEXTE
-
B I O D É G R A D AT I O N E N M I L I E U X P O R E U X
L’eau est un élément indispensable à toute forme de vie, et particulièrement aux
hommes. Les activités humaines requièrent déjà des quantités d’eau considérables
et, de manière évidente, la consommation a tendance à augmenter avec la croissance
de la population mondiale. Malheureusement, l’eau douce est relativement rare sur
Terre, et la quantité de produits chimiques qui la polluent est aussi positivement
corrélée à la taille de la population. Quelques exemples d’une telle pollution sont
illustrés ci-dessous :
• L’agriculture génère des quantités considérables de polluants organiques et
inorganiques tels que des insecticides, des herbicides, des nitrates ou des phosphates.
• Des produits chimiques, comme des solvants, des détergents, des métaux
lourds, des huiles qui sont présents sur des déchetteries illégales peuvent atteindre les réserves d’eau souterraine. Par exemple, de grandes quantités de
systèmes électroniques en provenance d’Europe ou des Etats-Unis sont envoyés en Afrique [228, 185]. Ces déchets y sont rarement traités et contiennent
des composés extrêmement toxiques (plomb, mercure, arsenique et autres).
Ces “e-waste” représentent une source de pollution importante et une menace
directe de santé publique [102].
• Certains médicaments se retrouvent en grande concentration dans les écosystèmes d’eau douce. Par exemple, les hormones présentes dans les pilules
contraceptives, en particulier les estrogènes, affectent profondément les poissons, menant à un phénomène de féminisation et à l’apparition d’espèces
“intersex”[136].
Par conséquent, la gestion des réserves en eau nécessite une attention particulière.
Comprendre comment se comporte les polluants dans les écosystèmes d’eau douce
est devenu une priorité. La zone contaminée, c’est-à-dire, le site de pollution original
et la zone qui a été touchée à travers des phénomènes de transport doit être clairement
207
208
identifiée parce-qu’elle nécessite un traitement particulier. Pour atteindre cet objectif,
il est nécessaire de :
1. comprendre les processus qui sont impliqués dans ces phénomènes de transport. En premier lieu, on peut imaginer que les molécules toxiques sont transportées dans les rivières et les lacs, mais il s’agit d’une image très imprécise.
L’eau douce liquide se trouve principalement dans les sols et les aquifères. Le
transport des polluants à travers ces formations géologiques met en jeu des
phénomènes complexes incluant de la diffusion moléculaire, des mouvements
de convection, des phénomènes de sorption, des réactions hétérogènes et de la
dispersion.
2. mettre au point et optimiser des techniques pour la récupération et le traitement de ces polluants. Par exemple, les polluants organiques peuvent être
dégradés par de nombreux microorganismes (endogènes ou exogènes). Ce
phénomène, que l’on nomme habituellement biodégradation, décrit la capacité des microbes, et des bactéries en particulier, à modifier la spéciation
chimique des éléments présents dans le milieu. Ces minuscules organismes
peuvent rompre les chaines moléculaires de composés toxiques pour produire
d’autres espèces chimiques. Quand les produits de cette réaction sont inertes
ou moins toxiques, les microbes peuvent être utilisés directement pour purifier
l’eau. Par exemple, les huiles pétrolières, composées de molécules aromatiques
toxiques, peuvent être dégradées par une classe de bactéries hydrocarbonoclastiques (HCB) [292]. La biodégradation peut aussi produire des composés
extrêmement toxiques et cela doit être pris soigneusement en considération.
Par exemple, la dégradation anaérobie du perchloroethylène (PCE) peut mener
à une production de trichloroethène (TCE), dichloroéthène (DCE) and chloride
de vinyl (VC).
Pour réaliser ces deux tâches, les phénomènes de transport doivent être décrits
quantitativement en utilisant une modélisation mathématique. Cette modélisation
requiert, en général, certaines simplifications. Dans ce travail de thèse, les polluants
sont traités en tant que solutés. Cela signifie qu’il y a toujours une partie de ces
polluants qui est dissoute dans l’eau, même relativement peu et que les processus
significatifs, en terme de transport, sont ceux associés à la portion miscible. Par exemple, les liquides denses formant une phase non-aqueuse (NAPLs) forment des blobs
persistants, piégés dans la matrice poreuse, qui se dissolvent peu à peu dans l’eau. De
plus, ces solutés seront considérés comme des traceurs (potentiellement réactifs). On
entend par là que le contaminant est présent en relativement faibles concentrations,
i.e., la densité et la viscosité de l’eau ne soit pas modifiés. C’est généralement le cas
pour une large quantité de polluants, pour les NAPLs par exemple.
On a aussi besoin de caractériser l’aquifère lui-même. Pour ce faire, on peut utiliser
certains traceurs non-réactifs. L’introduction de ces solutés dans les milieux aquatiques permet d’observer les courbes d’élution en effectuant des prélèvements à partir
D.2 C O N C L U S I O N S G É N É R A L E S
de puits répartis autour de la zone d’introduction. On peut en tirer des informations
concernant la direction de l’écoulement et l’organisation générale du milieu. Cependant, une interprétation correcte des ces données et des développements théoriques
de modèles de transport requièrent des informations sur la topologie du système à
une plus petite échelle. On peut, par exemple, utiliser la microtomographie à rayons X
pour obtenir une images en trois dimensions de la structure poreuse à petite échelle
sur des volumes relativement larges.
D.2
CONCLUSIONS GÉNÉRALES
Dans cette partie, nous proposons une synthèse des différentes conclusions propres
à chaque partie. Nous souhaitons tout particulièrement replacer ce travail dans le
cadre d’une stratégie plus globale. La section suivante est réservée aux différentes
perspectives, et aux travaux en cours de réalisation.
Dans cette thèse, nous nous sommes intéressés aux phénomènes de transport
dans des milieux poreux colonisés par du biofilm. Le travail présenté est basé sur des
analyses expérimentales, théoriques et numériques qui forment une base nouvelle
pour l’étude des ces systèmes complexes. Dans un premier temps, partie II, nous
présentons une technique d’imagerie du biofilm dans des milieux poreux opaques.
Cette avancée technique, développée entièrement durant ce travail de doctorat, utilise
la microtomographie à rayons X (ou tomodensitométrie) assistée par ordinateur, en
conjonction avec un choix spécifique d’agents de contraste, pour obtenir une image
en trois dimensions de la répartition spatiale des trois phases (biofilm, eau et solide)
avec une taille de voxel d’environ 9 µm. Cette approche s’avère très prometteuse pour
étudier la croissance de biofilms dans des structures poreuses ainsi que la réponse
des microorganismes à différents stimuli et conditions environnementales.
En utilisant cette information, on peut construire une stratégie de modélisation
pour le transport de solutés dans des milieux poreux colonisés par du biofilm. Dans la
partie I, nous présentons cette approche (MVMV) qui peut se décomposer en quatre
étapes :
1. imager la croissance de biofilm dans des structures poreuses et formuler les
équations qui décrivent les différents phénomènes à l’échelle du pore.
2. incorporer ces composantes dans des modèles efficaces à l’échelle du pore, par
exemple, en utilisant des formulations de type Lattice-Boltzmann (e.g., [116])
et valider la description mathématique proposée.
3. moyenner les équations différentielles à l’échelle du pore pour obtenir des
équations à l’échelle de Darcy (eg. parties III et IV) et calculer les propriétés
effectives associées à ces modèles sur la base de la géométrie 3-D capturée par
la technique d’imagerie.
209
210
4. développer les domaines de validité des différents modèles macroscopiques, et
valider l’analyse théorique par des mesures à l’échelle de Darcy.
Dans cette stratégie, les équations qui sont utilisées à la petite échelle sont validées
en comparant des résultats numériques directs (automates cellulaires ou modélisation cellulaire individuelle) avec les observations tri-dimensionnelles. De plus,
les modèles macroscopiques sont aussi étudiés et validés avec des expériences à
l’échelle de Darcy. Une telle approche, bien que fondamentale, est extrêmement
complexe à appliquer. Dans notre étude, on simplifie la démarche en considérant
une situation théorique pour laquelle le système d’équations décrivant les processus
à l’échelle du pore est bien connu. Sur cette base, on développe un théorie pour le
transport macroscopique de solutés non-réactifs et/ou biodégradés dans des milieux
poreux colonisés par des biofilms. Pour filtrer l’information à l’échelle du pore, on
utilise la méthode de prise de moyenne volumique, une nouvelle décomposition
en moyenne plus perturbation et des analyses en terme de moments spatiaux. De
nombreux modèles à l’échelle de Darcy sont présentés, ainsi que leurs domaines
de validité et des illustrations numériques. Si d’autres configurations décrites par
un jeu d’équations différentielles linéaires devaient être considérées, il serait tout
à fait possible d’adapter la théorie développée dans ce manuscrit. Les travaux en
cours incluent le développements d’outils numériques pour calculer les propriétés
effectives sur les images 3-D obtenues.
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Multiple-scale analysis of transport phenomena in porous media

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